Shimano shifters are notorious for this behavior.
The lubricant (grease) congeals to a sticky and stiff consistancy.

I agree with the hairdryer trick as a first step.

I personally use a citrus degreaser, sprayed into the shifter body and let it sit for a while.
Blow out with compressed air while running through the gears and then a light spray of lubricant (NOT WD40!).

rusty