What are the cons of implementing a 3 team multiplayer FPS game?
Well, apart from the other answers: What are the benefits?
As opposed to RTS, where you have a global strategy you can follow, in FPS you don't really have that much strategy, you see an enemy, you shoot, and a wounded teammate to help.
It doesn't really matter whether there is one enemy team or two.
The only benefits are: more targets to shoot at and enemys can be distracted from your team (which is a huge con if you are the distracted one)
But there is a 3 faction FPS: Planetside 2. (and 1 but that's been shut down)
However, it is an MMOFPS, not a pure FPS.
And that explains why you don't see 3 teams in regular FPS: because it's bad.
Coming from a Platoon leader standpoint there (meaning that I actually command 40 people on a 900+ player continent: The three-way fights (which are those where all 3 factions have significant population) are the worst.
They might be fun for farming (=racking up cheap kills) as an experienced player but for getting any objectives (capturing bases) and for newer players they are hell.
The only reason the 3 factions work in that game, is that these 3 way fights are the exception rather than the norm.
You have a giant map with over 800 players and usually 20 vs 20 (well from 1 on 1 to 96+ on both sides) fights between two factions on one base.
The 3-Faction-part is only relevant for the metagame that the platoon leaders play/ think about, but that is nothing else than RTS with your units being individual actual humans. Here it's important to have 3 sides: to prevent / limit imbalances, but again, that is metagame and thereby RTS territory, no direct gameplay impact for the FPS player.
On smaller single-map games the downsides would be even worse.
Objectives become almost uncaptureable and you get killed from random places, because EVERYWHERE can be where an enemy is.
Sounds like an incredibly frustrating experience.