(I'll try to provide a broad-strokes general answer. YMMV depending on specifics you haven't provided.)
For an existing codebase? Not many. Code written for C++11 and above can make use of a lot of new syntax and semantics and improve performance (e.g. with constexpr functions, move semantics and so on); in your case, that could be the standard library, or other libraries you're using which support a newer C++ language standard. But if the performance-critical code is your own, I would not advise promoting the standard version.
Having said that - if you were to revamp your actual codebase with C++11 (or better yet, C++14 or C++17 which are out), you would be able to benefit, in terms of performance, readability, maintainability, reduction of code size etc. That's definitely advisable, since more and more code will require C++11 or higher to compile, over time.