If you're into the whole Times Square/Da Club vibe for date night (or if you're going to see a show on Broadway and want to eat somewhere nearby), I remember R Lounge having some decent vegetarian options. Might have to talk to them in advance to get a window seat though, because it gets packed at night, especially on weekends. You'd mostly be there for the view, which is amazing, and it's one of the few places in the world where the view gets better when it rains.

If you eat some meat, just not a lot, Jeepney in the East Village has some great examples of Filipino food and a fun tropical bar vibe. Pancit usually has chicken or pork, but theirs can be made vegetarian if you ask. They also have balut and dinuguan (fetal duck eggs and pork blood stew), which are rare to find as menu items in the US.

If you like Indian food, Murray Hill has a huge range of options, and a lot of them are super affordable-- many have great lunch specials, and at night you'll see cabs parked all along Lexington Ave between 25th and 30th Streets because their drivers are all grabbing dinner. I hear Chennai Garden is open until 10pm. Ambience-wise it kind of teeters on the edge of hole in the wall, but you'll be too busy stuffing yourself anyway. And since it's within walking distance and you'll be there in July, afterwards you could amble over to Madison Square Park and get some milkshakes at the original Shake Shack, which is open until 11pm and has outdoor seating under cute market lights.