I think it depends on how many weeds there are on your paths. If its just the odd one a small flame gun would suffice but to be honest you might as well knock em up with a hoe. I use a push pull, two sided hoe mostly and find it quite quick and easy. If your paths are well covered in weeds it might be better to hire an industrial flame gun, (the type used to burn off road markings), but I know from experience that can be expensive. Personally I don`t think a flame gun is going to save any time in the long run, weeding is a case of little and often. Another solution might be to lay “woven fibre” weed suppressing membrane, I bought some off the internet recently to lay between some rows of newly planted raspberries.
Pete