Strictly speaking, though, there is no general answer. A CPU that does addition by incrementing a number n times is not unthinkable. It all depends on how the logic is implemented. For a proper answer, you need to check your CPU's documentation.
In the particular case of modern x86 implementations, ADD reg32,reg32 takes 1 clock regardless of the contents of src and dst.
If you're dealing with bignum classes, then most likely the + and += operators utilize the ++ operator in a loop. Although some are more mathematically beutiful and do it in a somewhat more efficient way. There are a lot of the other looped ++ bignum classes out there, though, because there are a lot of beginners trying to do it the simpler way. So, in the case of a bignum class, do like all teachers do and get to know your class! ;)
It may also be noteworthy to consider linked list classes, such as the std::list class, where with iterators, you can only use ++it or it++, there is no it + x or it += x operator for list iterators like there are for vector iterators. This leads to iterating being less effecient the further you have to iterate.
Modern computers operate on things other than bits (1s and 0s). They have circuits that allow them to operate on integers and real numbers. They essentially have circuitry to add any two numbers.