What gives off heat radiation?
Anything that has enough internal energy to do so.
Heat is simply the transfer of energy due to a difference in temperature. The way this is done is through infrared radiation - usually.
When an object has energy, it will emit this energy as waves of different lengths and frequencies. As the object gets hotter, the wavelength will get shorter and frequency higher.
Most objects on earth are at a temperature where they emit infrared radiation (a lower frequency wave) but really hot objects like the sun can emit high frequency radiation like ultraviolet.
Think of an electric heater. At first, you feel it get a little warm as it emits infrared radiation. Then, as it gets more energy from its power source, it starts to glow red. Now it is emitting infrared radiation and visible light.
So basically, any object that has heat or rather energy will emit some form of electromagnetic wave / radiation - but objects that are too cool (or low in energy) to emit visible light or high frequency radiation will emit just infrared (or heat radiation).
I hope this helps; let me know if I can do anything else:)