Several searches of NTRS and the FAA did not reveal a direct answer, so this is an educated guess.
The Kennedy Space Center Story does confirm that the tower was painted red during the Apollo era:
Each launcher is 445 feet tall and weighs about
12,000,000 pounds. The two-story base, enclosed by battleship grey
steel plates, covers half an acre. Within the base are computers linked
with the computers in the firing room of the Launch Control Center
and also connected with other launch related equipment.
Towering over the base is the red umbilical structure; it provides
support for nine swing arms for direct access to the space vehicle,
17 work platforms and distribution lines for propellant, pneumatic,
electrical and instrumentation systems.
This source used to be available on NTRS as document 19710024295, but seems to be no longer available. The similar book Moonport makes no mention of painting or color at all.
The mention of "battleship grey" suggests that it the very same paint that is used on naval ships. Such paint has a long history of being durable and corrosion-resistant -- an important factor for a facility near the ocean. Thus, it would be the expected default paint color at KSC.
I suspect that the red color of the tower is a kind of aircraft warning paint. FAA rules require such paint for structures taller than 200 feet, so they are easily visible by aircraft; the mobile launch platform is 445 feet tall. The rules do allow the FAA to grant exceptions; in particular, the MLP tower is solid red, instead of alternating stripes of "aircraft orange" and white. Also, even though KSC is in a restricted airspace, NASA and the Air Force do fly in the vicinity, so it is a prudent idea to increase the visibility of the tower.
The first two Shuttle stacks were painted entirely in white, including the external tanks. It would therefore be prudent to continue to paint the towers in a highly visible red color. However, the external tanks of all of the remaining missions were unpainted. Conveniently, the natural color of the external tank insulation was an orange color, and highly visible. Thus, there was no longer a need to paint the tower red and the more reliable grey paint could be used.