Can a jellyfish sting you even when it is dead?

I believe so; the sting is not (afaik) related to what the jelly has for a brain but is rather a more local chemical reaction.
A dried jellyfish would pose no threat.

Yes, they can definately sting when still dead, the nematocysts (the bit that stings you) works in a mechanical and chemical way, so touch fires them, it is not done under the control of the jellyfish. also if you have been stung do not rub it with your hand as there will also be nematocysts left on your skin which will fire both into your hand and more into your sting.
It is also not true that peeing on it will always help. for one thing it will not help the pain at all. Vinegar is good for box jelly stings as it hardens the nematocysts left on your wound and lidocaine antiseptic spray is good for stings from things such as man o war and sea nettles. treatment for jelly stings is very much species specific.

What an appropriate question for this site! Are you sure you’re not speaking metaphorically?

Yes.

This is off-topic but I could not help thinking of the line in the Bogart film To Have And Not Have: “Ever been stung by a dead bee?”