When you crack it open, you're looking for a tiny pawl, a little part about 6mm across, pivoting on a little pin about 1.5mm thick. It's actuated by a little hairspring, which is too weak to overcome the friction in the pivot when the grease goes gummy.

There's one attached to each lever, so it can fail in either or both directions. Depending on the model of shifter, one or the other might be easily visible or hiding under other bits of the mechanism. Once you find the pawl in question, the trick is to squirt some solvent at the pivot, and wiggle the pawl back and forth until the gunge in the pivot dissolves. You might need to use something sharp like a scribe to reach through the mechanism, and you'll have to push it in both directions until the spring can overcome the friction. It might take so long that you think it's not going to come good, but if you persist it's a guaranteed fix.

Flush the whole shifter with solvent, and make sure the other pawl is moving freely, then lube with a heavy oil instead of the original grease. After this you should never need to touch it again, because most Shimano shifters are actually extremely robust despite their complexity, and usually only ever fail due to the grease going hard.