The heat signature of what, the exhaust? That's very low density, so emissions from the exhaust itself will be negligible.
The plane itself will emit black-body radiation, but that is based on the skin temperature, not the temperature of any interior engine parts.
In the meantime, all of this is for passive detection, so at a distance you'd be asking for the detector to pick an at-best-tiny signal out of the entire sky. Moreover, clouds are opaque.