What happens to live animals in cargo depends on the type of airplane and the class of cargo compartment they're traveling in. For a typical passenger airliner, if you check your animal as baggage, they're going into the "lower lobe cargo compartment" under the cabin floor. This is classified as a Class C compartment, which essentially means that the crew have no access to it in flight, so it must contain fire detection and extinguishing systems. The applicable regulation is 14 CFR 25.857.
Quite simply, there are no considerations for live animals in Class C compartments. If you look in Boeing cargo fire checklists (copyrighted so can't post here, but you might get lucky with Google), the steps in the checklist deal with discharging the Halon into the cargo compartment and adjusting the pressurization system to minimize flow into the cargo compartment. This helps maintain the minimum required Halon concentration to provide the first big knockdown blast to stop the fire from spreading, and then maintain a low-level concentration to keep it suppressed for the length of the diversion to a landing. Unfortunately, that means any live animals will suffocate since Halon displaces oxygen.
For dedicated freighters such as the 747-400F/747-8F/777F etc., the main deck area is considered a Class E cargo compartment, which doesn't have a fire extinguishing system. If a fire is detected here, the procedure is to depressurize the entire cabin and minimize/cutoff air flow to the main deck area, and climb/descend to a mid-altitude (somewhere in the 24,000-28,000' range depending on the airplane) to minimize oxygen and help suppress the fire. If animals are transported here, rather than the Class C lower lobe compartments, they may survive. They may become hypoxic and lose consciousness, but there's no Halon to suffocate them. Depending on the length of time for the diversion to safe landing site, the lack of oxygen may be a serious issue for some animals just as it can be for humans, but that's a guess as I'm not a vet.
Bottom line is if you're taking your pet with you on a commercial flight, the safest, most comfortable, and least traumatic place for them is with you as a carry-on item.