First – heat

I've let myself cut out this part (and edit the other one), because I forgot a couple crucial things about thermodynamics, and made it really stupid. Sounded smart at the time, but it wasn't. There's just no good way to dispose of heat in space, only through radiation. Thanks for the guys for pointing out where I was wrong.

The other one – radiation

Everything glows, right, even if it's IR light, visible through thermals. That's important for combat, as we can see today. In space combat it's probably also important – remember, you don't die if you don't get hit, you don't get hit if you don't get seen, and you certainly can get spotted, when you use radar, not so much when you just observe through thermals.

How I'd deal with it? Simple – reflect or refract. The first one's simpler (yet as people explained to me, won't work, because it just trapps more heat inside, and then we die, but I'll leave it here, because maybe they have some other nuts technology in your setting, that may allow them to give the finger to thermodynamics), we can already do it with a mylar blanket – which is or can be used with good effect in war, cuz it appears to work (the issue's that it can work on Earth, because, due to having other means of dissipating thermal energy, it won't fry us). In a sci fi setting it can be done cooler, more advanced.

As for refraction – I got this idea when thinking about stealth suits (think Ghost in The Shell thermooptic camo). You use a material that refracts the thermal radiation you emmit outside the detectable spectrum (perhaps in some applications noise is needed, but that can be done). This works assuming the ones seeking your signature will look for the specific spectrum of EM radiation you should emmit from heat, so even if it has the same energy after getting refracted, the idea is it won't get picked up (unless they build sensors to counter that too, but that's not the point).

That's my point on those issues. I may be wrong, because, well, I don't have the education to understand it 100%, so I'm happy to hear your opinions on the topic, and corrections, if I'm wrong on something. Cheers.