Seems odd; when STIs fail they usually don't want to release the cable. What happens at the lever when you try to shift to the big ring? Does it move at all?
Ensure that the indexing isn't out by pulling on the cable and trying to shift to a ring below your granny ring. If the lever shifts and you can pull the cable a good distance from the frame, the fix is as simple as pulling more cable through your derailleur's pinch bolt.
However, I'm guessing there's a good chance that pushie's a bitsa and the lever isn't for a triple... if it shifts reliably through two positions, it should shift between three if it has them. Ensure it really is a triple shifter before... hang on. Deore is MTB gear, innit? Did Shimano use the STI acronym for that too?
So I guess that rules out a double lever... I'd say Rapidfire has pretty much the same guts as STI, meaning you may enjoy success by flooding the mechanism with as much degreaser as you can get in there. If it starts working, let it dry out some and lube it with heavy oil. If that doesn't do it, it's prolly junk unless you have what it takes to defy what is basically a 'no user-serviceable parts inside' situation.
It's a sticking spring-loaded pawl inside the housing. Seen it a hundred times.
Soak the internals with WD40 or such and give it time to penetrate. If it's still not working, you can remove the cable, unscrew the housing screw, remove the housing, and poke at the visible pawls around the toothed ring with a small flathead screwdriver. Make sure all of them are able to move easily and have springs that pull them to where they should be. Spray down any that stick and work them back and forth on their pivot until they're not sticking anymore.