This article spells out some reasons F#'s performance is occasionally better than C#. It says in it's "Firstly" section that only F# generates tail calls.
What exactly does that mean? And why is it a performance boost? This one thing may actually make or break between F# and C# for my chess app, which uses a ton of recursion.
Performance will depend more on the way you implement your program than the language. F# may generate IL better for some things while the C# compiler will be better for others. When choosing the languages you should consider other things rather than just performance.
If you're writing your chess program to learn F#, give it a try, it's an awesome language, just don't expect super blazing fast programs just because you're using a functional language.
Edit to answer the new question:
The F# compiler indeed does generate IL that has the tail. op code whareas the C# compiler doesn't. That by itself doesn't make F# faster or more performatic than C#, as you can see in my original answer above, but can indeed make a difference in your specific chess app, since you are stating that recursion is heavily used.
As a side note, the CLR may generate some simpler tail call optimizations during runtime, so for simpler functions in a x64 enviroment, even IL generated by the C# compiler may have tail call optimization.