I'd never come across these before, so was inspired to check them out, as Berkeman suggests there is some evidence that they can help and there is a folk belief about using hot spoons, it's not clear why they should work, but there are some theories. One suggests the heat disrupts some of the inflammatory proteins, but this seems unlikely, the heat from a 3-6 second application wouldn't go far and the effect on a range of bites and stings suggest a number of different biochemical pathways. You would need to apply the heat immediately after the bite, as the inflammatory proteins are releases almost immediately. Using temperature to alter blood flow might have some effect, and cold is commonly used as well as heat.
I did notice that some people comment that the sensation of heat is as painful as the bite, this makes the idea of the heat acting as a counter irritant altering the nerve conduction of pain, more credible, capsicum creams use the same principle. In this case, the nerve isn't deadened, our nerves use a variety of ways to sense and transmit pain signals, they also appear to prioritise some sensations as more dangerous, heat for example, this happens before any processing in the brain. Counter irritants can overstimulate some nerve receptors and modify the sensory route, but really the most effective are used over an area rather than a specific point. Unfortunately, all studies on pain are plagued by subjectivity, attention/context, memory and various other placebo influences. Along with these issues, many people describe alterations in the inflammatory responses, which these ideas couldn't adequately explain.
It's interesting, but the effectiveness seems very variable, and some people describe the sensation as worse than the original bite, currently I'm not really tempted to dash out and buy one, insect repellant works, well most of the time anyway. I suspect that if immunologists were convinced the initial inflammation could be controlled, there would be an interest in testing this as a way to prevent some anaphylactic reactions, in fact the only comments by an immunologist I found about these devices were lukewarm at best.