In textbooks, they say "absorb" heat and "give off" heat.
Is there a single word which can perfectly take the place of "give off"?
Emit (v): to throw or give off or out (as light or heat)
Radiate (v): to send out in or as if in rays
Emanate (v): to come out from a source - a sweet scent emanating from the blossoms
So while emit and radiate mean "give off", emanate means "be given off". A car engine would emit heat, but heat would emanate from the car engine.
Interesting to note: radiate can be used either transitively or intransitively, and can mean either emit or emanaate. What a flexible word!
As FumbleFingers points out:
radiate heat is even more common than emit heat. But I think that's because to some extent the word radiate actually means to emit electromagnetic radiation (which for most purposes means energy, or heat).
Physicists say "radiate" in reference to heat given off as thermal radiation (i.e. infrared heat like those heat lamps you see keeping food warm). They also use "convect" and "conduct", but these do not give the sense of direction that "radiate" or "give off" do.
"Emit" comes to mind. I also like "shed".
A chemist would say the opposite of absorb is desorb (the distinction is made, I guess, because the absorbed thing would be a chemical of some kind and it is more substantive than photons which are emitted or radiated).
If the thing being absorbed is a single unit, then "expel" would be the word to use.
At the moment, though, I can't think of anything outside of a sci-fi/fantasy context. Like in Animorphs #12, The Reaction, Rachel expelled a full-sized crocodile... Although for humor purposes, the word "burp" was used instead.