I did some extensive testing with movespeed a few weeks ago and no, it doesn't make you faster. I used livesplit to time a certain stretch on the training center mission and here's what I've found:

• Jumping doesn't affect speed (on flat ground anyways)
• I believe strafing and going backwards were both slower than moving forward
• Movement has 2 speeds when it comes to weapons: Slow and Fast.
• Fast movement is with the knife, your pistol, grenades and any sort of "lightweight" gadget.
• Slow movement is with any main rifle, submachine gun or rocket launcher
• Base movement speed is tied to a character.
• Females are faster than males, with Snakebite the fastest on land and Mako the fastest in water.
• The difference between the characters and their specialties is actually quite significant
• The LAR-V is absolutely useless on any character, does nothing.

Now, this isn't the pinnacle of testing, like I said, I just ran back and forth with a timer going and timed out the various scenario's.

In general Snakebite is the fastest character and should be fine for most if not all missions. I've not yet gotten any solid data on the training center though since you spend a good 40 something seconds in the water with Snakebite and it's 20 or 30 something for Mako. Maybe there's something there, but as a general rule of thumb: Snakebite is your girl and running around with nades, pistol or knife out is the fastest.