Can heat from the engine inlet be used to produce thrust?
I guess the heat you ask about is the temperature increase that goes with the precompression of air ahead of and in the intake. This heat is further increased in the compressor, according to the ideal gas law. If there were no temperature increase during compression, the engine could burn a lot more fuel and produce a lot more thrust.

To use the precompression heat would mean that first the air flows through a heat exchanger in order to transfer that heat to something that can produce thrust from it. That something would have to heat the air in the combustion chamber or after the last turbine stage in order to increase thrust. At both stations the gas temperature is a lot higher than the heat that can be harvested from the intake air. Now you first have to run that heat through a heat pump to make it useable (or, if you turn this around, you need a heat pump to produce a medium cold enough to harvest heat from the intake air in the first place!).

I hope you start to see how much more complicated and heavy this kind of engine will become. There have been attempts to recuperate the heat of the exhaust flow in turboprops, but so far nothing of that kind has ever entered production.