Recently, in an essay, I used "finishing up" in a sentence. My teacher marked it wrong, because to finish something isn't an action that takes time. But I've heard "finishing up" used in real life. Is it grammatically correct to use finish in present continuous tense?
Any action, including finishing something, takes a measurable amount of time between its start and its completion. It's not possible for any action to occur instantaneously, even if its length of time isn't perceptible. There will always be some moment before and some moment after—and some amount of time in between those two moments.
Further, if you're able to say that you're finishing up, then what you're doing is taking a perceptible amount of time—at least as long as it takes to say that, but most likely longer.
Your teacher may have been confusing finishing up with something like this:
I am finished.
That doesn't involve time, because finished in that sentence is an adjective, not a verb, and it's describing a property or attribute.
Context is everything of course so we can’t really tell if your usage was correct but the teacher’s answer seems unsatisfactory. Perhaps they were having a bad day. Finishing up us very common in spoken English, could you show us how you intended to use it in the written form?