Implementing any_of in C++
The algorithm looks correct.

Regarding the function signature, I'd make three changes:


You're not modifying elements, so take it by constant reference rather than reference.  
You don't need to specify a pointer to Function, Function is already a template parameter, and non-pointers could be valid (ex: a class with a () operator).   
"callback" isn't a very useful name for this case.  "predicate" would be better.   


While this form is fine, I'd also include a version that takes a start/end iterator instead of a container.

Also, there's no reason to include vector or iostream in any.hpp - only include things that are necessary.

For completeness in the testing method, you should also test with an empty collection, and a collection with no matches.