This doesn't determine "yes or no" to boost function. The performance drop may be well acceptable given program's requirements. More often than not, parts of a program are not performance-critical. And even then it may be acceptable. This is only something you can determine.
As to the standard library version, the standard only defines an interface. It is entirely up to individual implementations to make it work. I suppose a similar implementation to boost's function would be used.
This depends strongly if you are passing the function without binding any argument (does not allocate heap space) or not.
Also depends on other factors, but this is the main one.
It is true that you need something to compare against, you can't just simply say that it 'reduces overhead' compared to not using it at all, you need to compare it to using an alternative way to passing a function. And if you can just dispense of using it at all then it was not needed from the beginning
So, fear not. If your design/maintainability can improve from prefering std::function over virtual calls, try them. Personally, I really like the idea of not forcing interfaces and inheritance on clients of my classes.
There are, indeed, performance issues with std:function that must be taken into account whenever using it. The main strength of std::function, namely, its type-erasure mechanism, does not come for free, and we might (but not necessarily must) pay a price for that.
std::function is a template class that wraps callable types. However, it is not parametrized on the callable type itself but only on its return and argument types. The callable type is known only at construction time and, therefore, std::function cannot have a pre-declared member of this type to hold a copy of the object given to its constructor.
Roughly speaking (actually, things are more complicated than that) std::function can hold only a pointer to the object passed to its constructor, and this raises a lifetime issue. If the pointer points to an object whose lifetime is smaller than that of the std::function object, then the inner pointer will become dangling. To prevent this problem std::function might make a copy of the object on the heap through a call to operator new (or a custom allocator). The dynamic memory allocation is what people refer the most as a performance penalty implied by std::function.
I have recently written an article with more details and that explains how (and where) one can avoid paying the price of a memory allocation.