I watched the film and while the film is an absolute beauty, I couldn't help but think of the reasons why Elio and Oliver could not be together.
Since the movie explicitly doesn't mention any such reason, I can only assume that they weren't together because of social prejudices. However, Elio's family was pretty liberal: even before Elio had a conversation with his father towards the end of the film, a gay couple (friends of Elio's father) visited them, which shows that Elio's family wasn't really homophobic.
As of Oliver, he had a girlfriend back home, but he could have broken it off if he wanted Elio so badly. He implies at the end of the film, though, that his father would have put him in a "correctional facility", so for Oliver's father the romance would have been a strict no-no. This is the only valid reason I got for them to not be together.
So the reason boils down to--Oliver couldn't leave his parents for Elio. Is that it, or am I missing something? I found it really strange that while both Elio and Oliver were so into each other, they never raised a question on the imminent parting; they accepted it like it was a dogma.
I also noticed that Oliver seemed a bit dominant; after their first kiss, Oliver was the one hesitant to continue; yet he later gave in. He often was, or pretended, that he was indifferent to Elio; however, in one particular scene he asks Elio to come towards him, shows his affection (doing something) and then shuts the door! What was that about?