Upvote:7
Q: Is there a bag check or coatroom?
A: No, all coats and back packs must be carried while on tour. Please note that large bags and backpacks are not permitted and that there are currently no left luggage facilities available. For more information, see the Security page.
(emphasis mine)
http://visit.un.org/content/frequently-asked-questions
In response to the edit - there are plenty of commercial luggage storage options around New York City, so it depends on where you are staying (you may be able to leave it with your hotel concierge if you need to check out).
I won't list any commercial luggage storage options as I haven't used any in New York City, so I can't give you recommendations - but a quick search for "left luggage new york" gives several options within walking distance to the UN HQ.