How could this possibly cause a stack dive? More so, And what is
  RunContinuationsAsynchronously effectively going to do in order to
  solve that problem?
i3arnon provides a very good explanation of the reasons behind introducing RunContinuationsAsynchronously. My answer is rather orthogonal to his; in fact, I'm writing this for my own reference as well (I myself ain't gonna remember any subtleties of this in half a year from now :)
First of all, let's see how TaskCompletionSource's RunContinuationsAsynchronously option is different from Task.Run(() => tcs.SetResult(result)) or the likes. Let's try a simple console application:
using System;
using System.Threading;
using System.Threading.Tasks;
namespace ConsoleApplications
{
    class Program
    {
        static void Main(string[] args)
        {
            ThreadPool.SetMinThreads(100, 100);
            Console.WriteLine("start, " + new { System.Environment.CurrentManagedThreadId });
            var tcs = new TaskCompletionSource<bool>();
            // test ContinueWith-style continuations (TaskContinuationOptions.ExecuteSynchronously)
            ContinueWith(1, tcs.Task);
            ContinueWith(2, tcs.Task);
            ContinueWith(3, tcs.Task);
            // test await-style continuations
            ContinueAsync(4, tcs.Task);
            ContinueAsync(5, tcs.Task);
            ContinueAsync(6, tcs.Task);
            Task.Run(() =>
            {
                Console.WriteLine("before SetResult, " + new { System.Environment.CurrentManagedThreadId });
                tcs.TrySetResult(true);
                Thread.Sleep(10000);
            });
            Console.ReadLine();
        }
        // log
        static void Continuation(int id)
        {
            Console.WriteLine(new { continuation = id, System.Environment.CurrentManagedThreadId });
            Thread.Sleep(1000);
        }
        // await-style continuation
        static async Task ContinueAsync(int id, Task task)
        {
            await task.ConfigureAwait(false);
            Continuation(id);
        }
        // ContinueWith-style continuation
        static Task ContinueWith(int id, Task task)
        {
            return task.ContinueWith(
                t => Continuation(id),
                CancellationToken.None, TaskContinuationOptions.ExecuteSynchronously, TaskScheduler.Default);
        }
    }
}
Note how all continuations run synchronously on the same thread where TrySetResult has been called:
start, { CurrentManagedThreadId = 1 }
before SetResult, { CurrentManagedThreadId = 3 }
{ continuation = 1, CurrentManagedThreadId = 3 }
{ continuation = 2, CurrentManagedThreadId = 3 }
{ continuation = 3, CurrentManagedThreadId = 3 }
{ continuation = 4, CurrentManagedThreadId = 3 }
{ continuation = 5, CurrentManagedThreadId = 3 }
{ continuation = 6, CurrentManagedThreadId = 3 }
Now what if we don't want this to happen, and we want each continuation to run asynchronously (i.e., in parallel with other continuations and possibly on another thread, in the absence of any synchronization context)?
There's a trick that could do it for await-style continuations, by installing a fake temporary synchronization context (more details here):
public static class TaskExt
{
    class SimpleSynchronizationContext : SynchronizationContext
    {
        internal static readonly SimpleSynchronizationContext Instance = new SimpleSynchronizationContext();
    };
    public static void TrySetResult<TResult>(this TaskCompletionSource<TResult> @this, TResult result, bool asyncAwaitContinuations)
    {
        if (!asyncAwaitContinuations)
        {
            @this.TrySetResult(result);
            return;
        }
        var sc = SynchronizationContext.Current;
        SynchronizationContext.SetSynchronizationContext(SimpleSynchronizationContext.Instance);
        try
        {
            @this.TrySetResult(result);
        }
        finally
        {
            SynchronizationContext.SetSynchronizationContext(sc);
        }
    }
}
Now, using tcs.TrySetResult(true, asyncAwaitContinuations: true) in our test code:
start, { CurrentManagedThreadId = 1 }
before SetResult, { CurrentManagedThreadId = 3 }
{ continuation = 1, CurrentManagedThreadId = 3 }
{ continuation = 2, CurrentManagedThreadId = 3 }
{ continuation = 3, CurrentManagedThreadId = 3 }
{ continuation = 4, CurrentManagedThreadId = 4 }
{ continuation = 5, CurrentManagedThreadId = 5 }
{ continuation = 6, CurrentManagedThreadId = 6 }
Note how await continuations now run in parallel (albeit, still after all synchronous ContinueWith continuations).
This asyncAwaitContinuations: true logic is a hack and it works for await continuations only. The new RunContinuationsAsynchronously makes it work consistently for any kind of continuations, attached to TaskCompletionSource.Task.
Another nice aspect of RunContinuationsAsynchronously is that any await-style continuations scheduled to be resumed on specific synchronization context will run on that context asynchronously (using SynchronizationContext.Post, even if TCS.Task completes on the same context (unlike the current behavior of TCS.SetResult). ContinueWith-style continuations will be also be run asynchronously by their corresponding task schedulers (most often, TaskScheduler.Default or TaskScheduler.FromCurrentSynchronizationContext). They won't be inlined via TaskScheduler.TryExecuteTaskInline. I believe Stephen Toub has clarified that in the comments to his blog post, and it also can be seen here in CoreCLR's Task.cs.
Why should we be worrying about imposing asynchrony on all continuations?
I usually need it when I deal with async methods which execute cooperatively (co-routines).
A simple example is a pause-able asynchronous processing: one async process pauses/resumes the execution of another. Their execution workflow synchronizes at certain await points, and TaskCompletionSource is used for such kind of synchronization, directly or indirectly.
Below is some ready-to-play-with sample code which uses an adaptation of Stephen Toub's PauseTokenSource. Here, one async method StartAndControlWorkAsync starts and periodically pauses/resumes another async method, DoWorkAsync. Try changing asyncAwaitContinuations: true to asyncAwaitContinuations: false and see the logic being completely broken:
using System;
using System.Threading;
using System.Threading.Tasks;
namespace ConsoleApp
{
    class Program
    {
        static void Main()
        {
            StartAndControlWorkAsync(CancellationToken.None).Wait();
        }
        // Do some work which can be paused/resumed
        public static async Task DoWorkAsync(PauseToken pause, CancellationToken token)
        {
            try
            {
                var step = 0;
                while (true)
                {
                    token.ThrowIfCancellationRequested();
                    Console.WriteLine("Working, step: " + step++);
                    await Task.Delay(1000).ConfigureAwait(false);
                    Console.WriteLine("Before await pause.WaitForResumeAsync()");
                    await pause.WaitForResumeAsync();
                    Console.WriteLine("After await pause.WaitForResumeAsync()");
                }
            }
            catch (Exception e)
            {
                Console.WriteLine("Exception: {0}", e);
                throw;
            }
        }
        // Start DoWorkAsync and pause/resume it
        static async Task StartAndControlWorkAsync(CancellationToken token)
        {
            var pts = new PauseTokenSource();
            var task = DoWorkAsync(pts.Token, token);
            while (true)
            {
                token.ThrowIfCancellationRequested();
                Console.WriteLine("Press enter to pause...");
                Console.ReadLine();
                Console.WriteLine("Before pause requested");
                await pts.PauseAsync();
                Console.WriteLine("After pause requested, paused: " + pts.IsPaused);
                Console.WriteLine("Press enter to resume...");
                Console.ReadLine();
                Console.WriteLine("Before resume");
                pts.Resume();
                Console.WriteLine("After resume");
            }
        }
        // Based on Stephen Toub's PauseTokenSource
        // http://blogs.msdn.com/b/pfxteam/archive/2013/01/13/cooperatively-pausing-async-methods.aspx
        // the main difference is to make sure that when the consumer-side code - which requested the pause - continues, 
        // the producer-side code has already reached the paused (awaiting) state.
        // E.g. a media player "Pause" button is clicked, gets disabled, playback stops, 
        // and only then "Resume" button gets enabled
        public class PauseTokenSource
        {
            internal static readonly Task s_completedTask = Task.Delay(0);
            readonly object _lock = new Object();
            bool _paused = false;
            TaskCompletionSource<bool> _pauseResponseTcs;
            TaskCompletionSource<bool> _resumeRequestTcs;
            public PauseToken Token { get { return new PauseToken(this); } }
            public bool IsPaused
            {
                get
                {
                    lock (_lock)
                        return _paused;
                }
            }
            // request a resume
            public void Resume()
            {
                TaskCompletionSource<bool> resumeRequestTcs = null;
                lock (_lock)
                {
                    resumeRequestTcs = _resumeRequestTcs;
                    _resumeRequestTcs = null;
                    if (!_paused)
                        return;
                    _paused = false;
                }
                if (resumeRequestTcs != null)
                    resumeRequestTcs.TrySetResult(true, asyncAwaitContinuations: true);
            }
            // request a pause (completes when paused state confirmed)
            public Task PauseAsync()
            {
                Task responseTask = null;
                lock (_lock)
                {
                    if (_paused)
                        return _pauseResponseTcs.Task;
                    _paused = true;
                    _pauseResponseTcs = new TaskCompletionSource<bool>();
                    responseTask = _pauseResponseTcs.Task;
                    _resumeRequestTcs = null;
                }
                return responseTask;
            }
            // wait for resume request
            internal Task WaitForResumeAsync()
            {
                Task resumeTask = s_completedTask;
                TaskCompletionSource<bool> pauseResponseTcs = null;
                lock (_lock)
                {
                    if (!_paused)
                        return s_completedTask;
                    _resumeRequestTcs = new TaskCompletionSource<bool>();
                    resumeTask = _resumeRequestTcs.Task;
                    pauseResponseTcs = _pauseResponseTcs;
                    _pauseResponseTcs = null;
                }
                if (pauseResponseTcs != null)
                    pauseResponseTcs.TrySetResult(true, asyncAwaitContinuations: true);
                return resumeTask;
            }
        }
        // consumer side
        public struct PauseToken
        {
            readonly PauseTokenSource _source;
            public PauseToken(PauseTokenSource source) { _source = source; }
            public bool IsPaused { get { return _source != null && _source.IsPaused; } }
            public Task WaitForResumeAsync()
            {
                return IsPaused ?
                    _source.WaitForResumeAsync() :
                    PauseTokenSource.s_completedTask;
            }
        }
    }
    public static class TaskExt
    {
        class SimpleSynchronizationContext : SynchronizationContext
        {
            internal static readonly SimpleSynchronizationContext Instance = new SimpleSynchronizationContext();
        };
        public static void TrySetResult<TResult>(this TaskCompletionSource<TResult> @this, TResult result, bool asyncAwaitContinuations)
        {
            if (!asyncAwaitContinuations)
            {
                @this.TrySetResult(result);
                return;
            }
            var sc = SynchronizationContext.Current;
            SynchronizationContext.SetSynchronizationContext(SimpleSynchronizationContext.Instance);
            try
            {
                @this.TrySetResult(result);
            }
            finally
            {
                SynchronizationContext.SetSynchronizationContext(sc);
            }
        }
    }
}
I didn't want to use Task.Run(() => tcs.SetResult(result)) here, because it would be redundant to push continuations to ThreadPool when they're already scheduled to run asynchronously on a UI thread with a proper synchronization context. At the same time, if both StartAndControlWorkAsync and DoWorkAsync run on the same UI synchronization context, we'd also have a stack dive (if tcs.SetResult(result) is used without Task.Run or SynchronizationContext.Post wrapping).
Now, RunContinuationsAsynchronously is probably the best solution to this problem.