I would like to know if "the doors closed on him" is correct here:
He got in the elevator and pressed a button. Before the doors closed on him, she said "I love you".
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"close on somebody" means that the person is nearing something/someone that is already moving ahead of them.
Here, you are saying that the door is closing on him; but, he is already inside the elevator and not nearing the elevator. So, I would say, this is not correct, considering the context. It could be,
Before the doors closed on her, she said "I love you"
provided, she was rushing towards the elevator, as it was closing.