const correctness can’t improve performance because const_cast and mutable are in the language, and allow code to conformingly break the rules. This gets even worse in C++11, where your const data may e.g. be a pointer to a std::atomic, meaning the compiler has to respect changes made by other threads.
That said, it is trivial for the compiler to look at the code it generates and determine if it actually writes to a given variable, and apply optimizations accordingly.
That all said, const correctness is a good thing with respect to maintainability. Otherwise, clients of your class could break that class’s internal members. For instance, consider the standard std::string::c_str() — if it couldn’t return a const value, you’d be able to screw around with the internal buffer of the string!
Don’t use const for performance reasons. Use it for maintainability reasons.