Disclaimers: This isn’t medical advice. Don’t hurt yourself.

It’s only bad for you if you burn yourself. There was a study a couple of years ago showing a strong correlation between application of heat on insect bites and a reduction in percieved itchiness.

https://pmc.ncbi.nlm.nih.gov/articles/PMC10309056/

My pet theory is that two things are happening.

1. The mechanisms for sensing itch and heat from our skin overlap and can interfere with each other.
2. Transferring heat quickly enough breaks down the chemical or cocktail causing the itchiness. My best guess is that the heat denatures them ( https://en.wikipedia.org/wiki/Denaturation_(biochemistry) ) or the sudden shift provides a shock that’s disruptive enough to get the job done. Like moving something from hot, to cold, to hot again when sanitizing.

The linked study suggests my first conjecture is, at least, a reasonable explanation for the immediate relief we feel.

Itch and pain, distinct sensations with specific receptors and mediators involved, interact with each other. A painful stimulus, such as noxious heat, causes the inhibition of itch signalling.

There are a few recent studies floating about which address aspects of mosquito bite itchiness pathophysiology. We still don’t know exactly why that happens though. Without that info I can’t draw the same conclusion for my second conjecture.

Anecdotally I’ve found success treating mosquito bites by

1. Heating water while putting something cool on the bite
2. Once the water is near boiling: taking the water off the heat and letting it cool to ~60° C / 140° F
3. Soaking a bit of metal in the hot water until the metal heats up
4. Touching that hot metal to the bite for a few seconds.

A fork or knife works fine. Anything that will radiate the heat away from itself and into the skin quickly. Ice or cold water works for the cooling bit. I’ve also used an ice cold metal can a few times.

If the metal is too hot it can cause a burn. antiscald.com has a nice chart for this: https://antiscald.com/index.php?route=information%2Finformation&information_id=15

If we reference that table alongside the Denaturation wiki article’s chart we can identify the line we’re towing. We want something hot enough to kill the itchiness but not so hot it’ll damage our skin. 55° C applied to the mosquito bite for five seconds or so seems to do the trick. Your mileage may vary.