You can get relays with timers and just set it for however much after-run you want. They're available with different amounts of adjustability from micro seconds to minutes, maybe hours. I haven't paid a lot of attention to details, but I know they exist.
There's also small thermoswitches you could use that are available in different temperatures. The ones I'm thinking of are about the size of a quarter but thicker. I don't know what they're called. If you look up RV heater parts you'll likely see what I'm thinking of. They're used specifically for what you want, after-run of the fan to cool the unit after the fire has gone out. One of those in the right temp would turn the fan on and off with temperature, all automagically. The amperage rating is pretty small so it would need a relay to the fan.