Flamer robots
Real Life examples : Alcoholic Stepfather, B.O.B.
Game examples : B.O.B. replica in DSL 2.1, none in stock
There are numerous ways to use Flamethrowers on robots. Here is a list of the most common uses.
On SnS
Flamethrowers on SnS are not very efficient combat-wise, but do create an awesome "circle of fire" effect. As JoeBlo discovered, they can be put on flails (also valuable for HS) to greatly increase their range, as well as lag and Havok explosion potential. Also one of the most common uses of them, roughly equal in number to the pusher flamethrowers.
On HS
A similar approach to the flamethrower SnS, but with added cool looks - now the circle of fire can move.
On Pinning robots
Probably the most efficient way of using a flamethrower in combat is that to keep them aimed at an immobile enemy, and that is the job of a pinning robot. They have been seen on Rammers, Pushers, Wammers (probably the most efficient of these since the wedge helps at immobilizing the opponent), but one Juggler (built by 090901) has been seen using flamethrowers as well. Naryar is the specialist here.
On FS
The only example of flamethrower FS is JoeBlo's Volcano, an unique design (if not for an generic name) using flails to propel flames at a great distance forward.