There are many great ways to move furniture in NYC. The best way is to not move it, that is to leave it in the apartment for the next tenant to use. This benefits everyone, but is not always feasable because management companies are notoriously annoying about connecting you to the future tenant.
The second best way is to sell the furniture on craigslist. All you have to do is register your e-mail address and put up a free ad for all cheapies to see. The cheapies then e-mail you, arrange a time to pick up the item, and then not show up. Or they show up and give you $9 for your awesome futon. You think about how long it took you to get that futon into the apartment, assemble it together, wash the stains off of it, move it out, etc. Then you hold the $9 in your hand and silently curse Craig the Hippie.
So instead of hiring movers or begging strange Yugoslavian men to lift your dresser, I suggest you start a yard sale. First, empty your apartment of all the stuff you kind of like but don’t have the energy or space to move. Place it outside of your building. Stand on the stoop and watch strangers grab your stuff like it’s San Francisco in 1849. They’ll even carry a shelf to a 6th floor walk up, just because it’s free! Profit: zero. Entertainment: priceless.