tenant keeps personal belongings in stairwell

It's against your building's rules and against fire code.
Notify the tenant that they are not in compliance with N.Y.C. fire code requirements and managment (if you have a managing agent or do it yourself if you own the building) will remove the items if the tenant does not do so within 24 hours of the notification.

or, are you a neighbor? Where's the managing agent, they should have done that by now.

What do you mean by "remove"? Would I have to store his item or can I just throw it out? What if it has value? Can they sue?

O.K. you are the landlord.
Most tenants who get a request from the landlord to remove potential safety hazards from common ares would do it if they are informed of the consequences.

You have requested repeatedly (by e-mail I hope).
No compliance from the tenant.
So you e-mail them on Tuesday , the next business day after this holiday weekend.
If they say they won't be back within 24 hours, explain that you can not allow such a safety hazard to exist on your property.

Then gather the items and if you have room you can store them in the basement for a few more days. ( Not required and may send the message that you are a pushover. If you want to give them one more chance that's up to you but make it clear, in writing, that there will be no more chances or repeated requests. They are lucky you have been so patient. Circulate a memo to all tenants regarding such violations. In a smaller building you will be seen as a compassionate landlord but not a pushover.)

If they don't return to claim them, you can donate any items in the stairwell to charity.
If they were so valuable to the tenant, he would not be leaving those things in the stairwell.

Take photos of the items in the stairwell before you remove them. Send the photos attached with your e-mail.

You are the landlord and will be liable for not making certain the stairwell is clear if there is an accident or heaven forbid -- a fire.

There are several small building N.Y.C. LLs telling me that they would e-mail the tenant today and remove the items if they don't by 6pm. But that's only the LLs that have a janitor on duty today to do it for them. None is willing to drive back to the city from L.I. to do it themselves today.

Here's a list of some of the things they have had to remove from the stairwells of their buildings:
beach chairs
umbrellas
boogie board
empty suitcases
old sneakers and footwear of various descriptions
stepstools
brooms/mops/buckets
old rugs
old broken toys/ small appliances

It's up to you if you want to remove it today or wait until tomorrow to deal with it.

While you're at it, put up signs with your rules on the landings of each stairwell.
On the wall where it can be seen.