The manual says I can move or run faster if my "weapon is holstered" and says to hit the right mouse button. But that doesn't do anything except make the picture jump forward a little for a split second. Is there a keyboard key or combination to put your weapon away? Thanks.

Press and hold the reload button.

Dunno why it says to hit the right mouse button, that makes you aim - increases accuracy and uses a scope if the gun has one - but you go slower, not faster. b3w4r3 gave the right answer - press and hold reload and it'll holster in about a second or two.