Three things wrong with that circuit.
1) It only has a gain of 2, see how to calculate the gain of a non inverting op amp here:- http://en.wikipedia.org/wiki/Operational_amplifier
2) You in effect have a floating input, that is the input is very high impedance and is picking up all sorts of interference.
3) That sort of circuit will only amplify voltages, you are in effect putting a variable resistance on the input you are not inputting a voltage.
Try this. +ve input of amp connected to two 1M resistors, one goes to +5 and the other ground. Then feed this point and ground at your flame.
Change the resistor that goes from the -ve input to ground to a 10K. What sort of op amp are you using.