What you're asking about is essentially a Class E compartment. See this other question/answer for more info: Is the life of pets considered when dealing with a cargo fire indication?
Short answer, yes you can manually depressurize the cabin to try to suppress the fire. But it's a matter of effectiveness. The Class E compartments are used for freighters where the normal "cabin" area is carrying cargo, not passengers. There are three problems with using a Class E method for suppressing a fire in the cabin of a passenger airplane.
First, the passengers need oxygen. The oxygen masks given to passengers were only designed to give them oxygen in a concentration and duration long enough to keep them alive in the event of a depressurization, and the several minutes it'll take to make an emergency descent to 10,000' where supplemental oxygen isn't needed. The supply of oxygen isn't sufficient for a hours long, or even 30 minute long diversion in the event of a fire/smoke situation. Even if the decision was made to allow the passengers to become hypoxic by not providing them any oxygen because of the fire, that's not possible because the masks will drop automatically once the cabin altitude exceeds 14,000'.
Second, those cheap "dixie cup" masks are not fully sealing. If you look at any oxygen mask that the flight crew uses for example (http://aviationoxygen.com/aviation-masks.html), you'll see the difference between those and the yellow passenger masks. The flight crew ones are designed to fully seal the mask around the pilot's face so that the pure oxygen being supplied to the pilot doesn't leak into the surrounding environment where there may be a fire. Fire + pure oxygen = bigger fire. So the last thing you would want is the passenger oxygen system to be activated if there's an active fire in the cabin. This, by the way, is the reason why you won't see commercial pilots with beards or extremely long mustaches. Significant amounts of facial hair interfere with the sealing capability of those masks.
Third, and probably biggest reason, is that the Class E method of depressurizing does not extinguish the fire, it simply suppresses it. This has been proven in a couple recent accidents, most notably the UPS6 747-400 that crashed in Dubai (https://en.wikipedia.org/wiki/UPS_Airlines_Flight_6). Boeing and the industry as a whole, has been re-evaluating the (lack of) effectiveness of depressurization as a fire protection method, not to mention the carriage of lithium ion batteries.
The most effective way to deal with a fire in the cabin is to attack it directly with fire extinguishers, even if that means cutting into the interior structure and paneling to access the fire. In general, flight crew are provided with the tools and training to do that. The quality and extent of the training provided obviously varies widely depending on the airline, regulatory authority, country, culture, etc.