A paper review I've undertaken turning out to be harder than expected
Is it common for a reviewer to review only a part of the paper, and leave the rest of the paper to other reviewers?


I have done that on at least two occasions. Sometimes it is the only honest option.


  On the other hand: is it my duty, as a reviewer, to complete my review, regardless of how much time it takes?


Not really. It is commonly considered a duty of a career mathematician to contribute to peer review of mathematical works (at least) proportionally to their own publishing. Whether you achieve this by refereeing some really tough papers or a lot of simple ones is up to you.