This was also driving me crazy tonight.  I created a ToolTip subclass to handle the issue.  For me, on .NET 4.0, the ToolTip.StaysOpen property is not "really" stays open.  
In the class below, use the new property ToolTipEx.IsReallyOpen, instead of property ToolTip.IsOpen.  You will get the control you want.  Via the Debug.Print() call, you can watch in the debugger Output window just how many times this.IsOpen = false is called!  So much for StaysOpen, or should I say "StaysOpen"?  Enjoy.
public class ToolTipEx : ToolTip
{
    static ToolTipEx()
    {
        IsReallyOpenProperty =
            DependencyProperty.Register(
                "IsReallyOpen",
                typeof(bool),
                typeof(ToolTipEx),
                new FrameworkPropertyMetadata(
                    defaultValue: false,
                    flags: FrameworkPropertyMetadataOptions.None,
                    propertyChangedCallback: StaticOnIsReallyOpenedChanged));
    }
    public static readonly DependencyProperty IsReallyOpenProperty;
    protected static void StaticOnIsReallyOpenedChanged(
        DependencyObject o, DependencyPropertyChangedEventArgs e)
    {
        ToolTipEx self = (ToolTipEx)o;
        self.OnIsReallyOpenedChanged((bool)e.OldValue, (bool)e.NewValue);
    }
    protected void OnIsReallyOpenedChanged(bool oldValue, bool newValue)
    {
        this.IsOpen = newValue;
    }
    public bool IsReallyOpen
    {
        get
        {
            bool b = (bool)this.GetValue(IsReallyOpenProperty);
            return b;
        }
        set { this.SetValue(IsReallyOpenProperty, value); }
    }
    protected override void OnClosed(RoutedEventArgs e)
    {
        System.Diagnostics.Debug.Print(String.Format(
            "OnClosed: IsReallyOpen: {0}, StaysOpen: {1}", this.IsReallyOpen, this.StaysOpen));
        if (this.IsReallyOpen && this.StaysOpen)
        {
            e.Handled = true;
            // We cannot set this.IsOpen directly here.  Instead, send an event asynchronously.
            // DispatcherPriority.Send is the highest priority possible.
            Dispatcher.CurrentDispatcher.BeginInvoke(
                (Action)(() => this.IsOpen = true),
                DispatcherPriority.Send);
        }
        else
        {
            base.OnClosed(e);
        }
    }
}
Small rant: Why didn't Microsoft make DependencyProperty properties (getters/setters) virtual so we can accept/reject/adjust changes in subclasses?  Or make a virtual OnXYZPropertyChanged for each and every DependencyProperty?  Ugh.
---Edit---
My solution above looks weird in the XAML editor -- the tooltip is always showing, blocking some text in Visual Studio!
Here is a better way to solve this problem:
Some XAML:
<!-- Need to add this at top of your XAML file:
     xmlns:System="clr-namespace:System;assembly=mscorlib"
-->
<ToolTip StaysOpen="True" Placement="Bottom" HorizontalOffset="10"
        ToolTipService.InitialShowDelay="0" ToolTipService.BetweenShowDelay="0"
        ToolTipService.ShowDuration="{x:Static Member=System:Int32.MaxValue}"
>This is my tooltip text.</ToolTip>
Some code:
// Alternatively, you can attach an event listener to FrameworkElement.Loaded
public override void OnApplyTemplate()
{
    base.OnApplyTemplate();
    // Be gentle here: If someone creates a (future) subclass or changes your control template,
    // you might not have tooltip anymore.
    ToolTip toolTip = this.ToolTip as ToolTip;
    if (null != toolTip)
    {
        // If I don't set this explicitly, placement is strange.
        toolTip.PlacementTarget = this;
        toolTip.Closed += new RoutedEventHandler(OnToolTipClosed);
    }
}
protected void OnToolTipClosed(object sender, RoutedEventArgs e)
{
    // You may want to add additional focus-related tests here.
    if (this.IsKeyboardFocusWithin)
    {
        // We cannot set this.IsOpen directly here.  Instead, send an event asynchronously.
        // DispatcherPriority.Send is the highest priority possible.
        Dispatcher.CurrentDispatcher.BeginInvoke(
            (Action)delegate
                {
                    // Again: Be gentle when using this.ToolTip.
                    ToolTip toolTip = this.ToolTip as ToolTip;
                    if (null != toolTip)
                    {
                        toolTip.IsOpen = true;
                    }
                },
            DispatcherPriority.Send);
    }
}
Conclusion: Something is different about classes ToolTip and ContextMenu.  Both have "service" classes, like ToolTipService and ContextMenuService, that manage certain properties, and both use Popup as a "secret" parent control during display.  Finally, I noticed ALL the XAML ToolTip examples on the Web do not use class ToolTip directly.  Instead, they embed a StackPanel with TextBlocks.  Things that make you say: "hmmm..."