How do I implement a thermal camera?

the most common method is to just texture the world in thermal camera colors, render it into a FBO(or similar) and then perhaps add a few filters on top of that(noise, blur and maybe false color mapping).
any other effects you may see in a game are just a few clever hacks, like particle systems to simulate heat around hot objects or a creative use of vertex color to simulate heat spreading around surfaces near a hot object, or maybe render things slightly transparent to a few hotter objects, like people (the latest rainbow six games use this effect pretty often).

There is no correct way of doing this, so just be a bit creative.