If you will google "Flipper stays in up position", you'll find 975,000 hits.

Having said that, first decide if it is mechanical or electrical. When it stays up, turn the game off. If it drops, it's electrical, if not, it's mechanical.

In no particular order...pull up (away from the playfield) on each flipper. There should be some play, maybe 1/16" up and down. If one doesn't have the play, it will bind. Look at your flipper coil wrappers. If you see a burned spot, the coil may have overheated and burned the flipper coil sleeve. Move the flippers up at the same time, and feel for a difference. If one has more resistance than the other, and it still has up and down play, rebuild both flippers (get your kit from Pinball Resource and tell Steve I said hi. It also might be that the spring on the problem flipper is the wrong one or might be stretched or attached incorrectly, so inspect them and compare. That should keep you busy for a minute

I'd start by replacing the nylon coil sleeve and cleaning the plunger (metal part that goes in the coil sleeve). Cheap & easy potential fix.