Let's find out, courtesy of some clever sums by Vlad Tretiak.
(top image by Segaman)
Picking apart Valve's Source engine, and translating distance in the engine into actual real-world units of measurement, he's been able to determine both the "standard constant movement speed" and "maximum possible movement speed" (ie, using items or bonuses) of some of the most popular first-person shooter games on the planet.
The results are below. Before you ask "why isn't [other game] on the list?", remember, the project only used Source Engine games so that it could fairly compare the speeds; as Tretiak says in the footnotes to his results, "comparing dimensions in different engines is rough".
Nevertheless, he does squeeze a few other comparisons in, from UT2004, Quake III and Halo. No surprise, then, that Quake III comes away as the clear winner in constant speed, at least among those included.
If you're after the most obvious omission in a discussion about FPS speed — Doom Guy — PinkLionThing has done some isolated calculations and found that, if he'd been included on the list, he would have whooped even Quake II's arse, as he could travel 22 m/s, more than double the speed of his nearest competitor.