What happens if you touch a bioluminescent animal?

joke answer: They make a cute squeak! (:

real answer: Nah, they won't do anything. Bioluminescence isn't heat; it's an internal chemical reaction that produces light! :)

Would somebody actually imagine there's an animal that's glowing yellow to white hot and still alive?

Nothing. Bioluminescent animals are also made of meat like we are, if they release too much heat, they would cook themselves. Most bioluminescence are from the breakdown of luciferin by a luciferase and the reaction tends to be very efficient, so very little waste heat is produced. We very commonly use bioluminescence for experiments, I have personally done a lot of experiments with genetically modified flies that would produce bioluminescence upon feeding of luciferin

tl;dr: no

Bioluminescent animals don't make light the same way a light bulb does by heating something so hot that it glows. If they did, they'd be the first to burn wouldn't they. The light comes from chemical reactions that don't emit that amount of heat, if they emit any at all. Of course assuming the reactions are endergonic (which I think is a safe assumption) they have to be powered by other chemical reactions that do emit heat... But that's part of the general functioning of the organism; all living organisms emit heat but it's usually in amounts that don't even make them feel warm to the touch to us. In theory the bioluminescent area of the organism wouldn't even need to be warmer than other areas of the organism.

I can look up the actual thermodynamics if you're interested in a more precise answer. It really depends on the chemical reactions in question (which in turns will vary from organism to organism, bioluminescence evolved convergently many times) whether they do emit heat, how much, and whether it's to an extent we'd notice touching them. My intuition would say yes, not a lot, and no. But it's been a long time since I've touched a firefly.