How can lime juice 'cook' raw fish without any heat?

Because the firming that happens when food cooks is denaturation, the unfolding of proteins, and heat is only one way to cause it. The cold appetizer ceviche is prepared by chemically "cooking" raw fish and shellfish in an acidic citrus marinade, with no heat at all.

Denaturation happens when proteins lose the folded shape of their native state. The trigger can be heat, but also a strong acid or base, a concentrated salt, an organic solvent, agitation or radiation. A protein has to be folded into its native shape to work, and the hydrogen bonds that help hold that shape are weak, so they are easily affected by heat and by acidity alike. Acid-induced unfolding often happens between pH 2 and 5.

Eggs show the idea best. Fresh from the shell, egg whites are transparent and liquid; cooking turns them opaque and solid, which is why boiled eggs go hard and cooked meat goes firm. The same transformation can be done with a chemical instead of heat: poured into acetone, egg whites turn translucent and solid. Acid works on milk too: add it, and the protein casein denatures and coagulates into curds, the first step in making cheese. A citrus marinade does the same kind of job on the proteins of raw fish, which is why ceviche counts as "cooked" without ever touching a stove.