I've seen Limbsaver products "melt", though not specifically recoil pads. I've had no such issues with Kick-Eze pads.(?) It is a good idea to store any soft pad muzzle down, otherwise they will start to stay compressed. That's true of any recoil pad really, but obviously the really soft ones show it sooner.
Strictly speaking, any wood stocked firearm is better stored muzzle down, keeping oils and solvents out of the wood. (of course you have to tale precautions with the crown/muzzle being damaged or rusting.) Even though I have a couple old rifles that have suffered this fate, some horribly so ruing the stock at the wrist, I generally don't store them muzzle down, just a couple with thick soft pads.