As a first check of the Arduino analog input, replace the flame sensor with another 10K resistor. The analog input should read about 512 and be very steady. If not, you have a problem. Are all the grounds connected together?
The "flame sensor" is just an IR photodiode and they can be damaged by static electricity. If you have a multimeter, try measuring the resistance of the photodiode in both directions. It should be low in one direction and very high (but depending on local IR sources) in the other. If it tests as an open circuit, it has been destroyed.