The decision to have 4 firing rooms was made in the early 1960s, before the Apollo missions (e.g. the Lunar Orbit Rendezvous plan) were worked out. At the time, some of the options included multiple launches to assemble the lunar expedition in space. So they built a complex that could accommodate any conceivable mission structure.
The Launch Control Center for Launch Complex 39 is attached to the Vehicle Assembly Building (VAB). It originally was designed with four Firing Rooms, one for each High Bay in the VAB. Up to four Saturn rockets could be stacked in the VAB at the same time, and each would have a dedicated Firing Room for test, checkout, and launch.
So a firing room would be in use from stacking to launch of a mission. This makes it likely that more rooms were used simultaneously.
I seem to remember an aerial photo of Cape Canaveral with two Saturns on two launch pads. Can't find it now, but that would have been an occasion when two rooms were in use at once.
In the end, 3 rooms were finished as control rooms during the Apollo program. Firing Room 4 was used as the project management room for construction of LC-39 and the Apollo systems integration.
Rocket Ranch: The Nuts and Bolts of the Apollo Moon Program at Kennedy Space Center has a diagram that which room was used for which mission.
During the Shuttle era, 2 firing rooms were used simultaneously:
... firing room 2, which is still operational as a space shuttle support facility. During a shuttle launch and a mission, NASA will staff the room with several managers and chief engineers, but will not place any command and control people there. Essentially, it seemed like the place for managers to observe the mission while the folks in Houston run it.
However, there are two other firing rooms, Nos. 3 and 4, where the prime shuttle operations are run out of KSC.
These days, Firing room 4 has been divided into 4 smaller rooms (page 25).