I think any of the suggestions above for a verb plus competition (e.g. made it into a competition) would be fine and expected. However, if you really want a single word, there are a couple of existing words that might work.
Rivalize may be closest; it can mean something quite similar to what you describe. From Oxford Dictionaries:
rivalize (also rivalise)
VERB
no object With in, with. To enter into rivalry; to compete.
If your firefighters have rivalized, then they have entered into rivalry and begun to compete (with their fellow firefighters), thus turning a neutral exercise into a competition.
The purpose of the firefighters' training exercise was to learn a new
skill, but they (soon) rivalized.
Although the term is generally not used with an object, you could also stretch the term to rivalized the exercise and I think would be well understood.
Finally, adversarialization is a term that sees some use in a legal context1, and that might suit your purposes.
The purpose of the firefighters' training exercise was to learn a new
skill, but they adversarialized (it).
This term typically refers to "getting lawyers involved"—i.e., moving a discussion, negotiation, or dispute into the adversarial system. But it seems that it would work for any situation where a once-non-adversarial activity becomes adversarial.
1 For example,
The principal complaints center upon the adversarialization of the
proceedings that proceeds from growing lawyer participation in the
process.
(Thomas E. Carbonneau, Arbitral Justice: The Demise of Due Process in American Law, 70 Tul. L. Rev. 1945, 1959 n.42)