This warning is quite clear - you're starting an asynchronous task, but you have no mechanism for being notified when it's complete. What if you do need to know when it's done, later? Before returning from your method, or at a point where you need to be sure that the data is synchronized with the DB?
awaiting the operation is one way to ensure that, but as you say, you don't want to stop the independent processing while the data is saved. await is for asynchrony, not for parallelism.
Additionally, awaiting the Save task ensures that you catch any exceptions that are thrown by the DB. When you simply release the task, any exceptions are lost - at best, you can handle TaskScheduler.UnobservedTaskException to handle unobserved tasks that throw exceptions, but a much simpler way is to simply observe the task.
One way to observe the task is await, as you know. Another is to simply save the Task object that SaveChangesAsync returns in a variable. This will allow you to await (or Wait()) on that Task later, catch its exceptions (either by try/catching the await call or by attaching a ContinueWith to the Task) and will let the compiler rest easy knowing you're not abandoning the task to its fate:
//a lot of database dependant processing
var dbSaveTask = db.SaveChangesAsync(); //asynchronouly saving database changes
//a lot of ANOTHER long term database INDEPENDANT stuff
// now, before assuming the DB is up to date, ensure the TAsk is complete.
await dbSaveTask;
// now do something Db-dependent!
