If by detrimental you mean would it reduce your chances of admission, yes. Not because they will be “upset” with you, but because nyu is one of the schools where there is a clear advantage to applying ED. Also, judging from what my daughter (current freshman) has said, there seems to be a significant number* in the RD pool comprising those that were rejected or deferred from early ivy/T20 applications, implying that the pool itself is just more competitive than ED.
*actually tbh I don’t know that there is a significant number in the pool itself but she says a number of her classmates fall into this category, with excellent stats.
All that said, as @“Erin’s Dad” above has said, if you need to move the app for financial reasons, or even if you are just second guessing whether you want to be bound there if made an offer, then call admissions and tell them you need to move it. You don’t need to say why.