Why not try the real Finnish sauna (please pronounce as fauna) wherein the temperature of 80 to 100 degrees centigrade and additionally throwing hot water on hot rocks of the sauna stove readily cures itching from mosquito bites in Lappland (Northern Finland). Should also work on jellyfish sting pain.

Awesome!!! I will have to remember this. I've never been stung but I've discovered myself by jellyfish a few times.

How hot, painfully hot? Hot tub hot?

An inexpensive and reasonably convenient option for beach-goers would be to carry a solar shower device to the beach, filled and ready to heat in the sun. It took moments to find one for under US$15 on eBay. In short order, the black plastic bag will reach useful temperatures. If not needed for yourself or another inflicted beach-goer, it's a convenient way to wash off the ocean as you leave.

Works for Portuguese man of war stings, too. "Mo' Hotta, Mo' Betta!"

Whoa there, 100 degrees Centigrade is 212 degrees Fahrenheit! My sauna won't get that hot - thank goodness. The biggest problem I see for myself is that a jellyfish sting would likely happen while in the tropics snorkeling . . . and there aren't many saunas available there. :-) But I'll remember HOT packs as a treatment if the jellyfish sting happens.

Hot water feels jus as good as scratching on poison ivy and doesn't cause bleeding.

This seems like old news. For some time now, it has been known that insect stings (venom) are thermolabile, meaning they break down with heat. In fact there is commercial product based on this. My wife and I have been using a device called Therapik to treat wasp stings for at least 10 years now. I just checked their website
http://www.therapik.com/Therapik_more_information.htm
and found that they list a variety of venoms that can be treated with heat - including jellyfish. As for the methods employed to determine whether hot or cold is best, it might have made more sense to obtain samples of the venom and see if they break down when heated or cooled.

Instead of all these ideas about solar showers (not useful on cloudy days) and saunas (not too many saunas around the beach and besides which, hot, steamy air doesn't have anywhere near the heat content of hot water), you could just carry one or more of the chemical hand warmers if you expect to be in waters that may have jellyfish. Either the iron/air hand warmers or the reusable sodium acetate hand warmers should be able to quickly reach the temperatures stated in the article.