Tactical Operations: Advanced Rules, pg 16, allows a Mech to Sprint. While doing is it may not make any deliberate, Damaging Attacks, Spot for Indirect Fire, or any other action that would normally require it sacrifice an Attack.
Total Warfare, pg 224-225 covers Infantry Dismounting from a transport unit. It says that the infantry unit may dismount the carrier during the Movement Phase at the end of the carrier's move. It also says a charging carrier cannot dismount Infantry.
It seems like no rule covers an occasion where a Mechanized Battle Armor unit on an Omnimech, say Elementals on a Clan Fire Moth, doing a Sprint can Dismount or not. The interactions with Dismounting and Charging do raise some questions regarding this interaction, though, as the restrictions on the carrier are very similar for Sprinting and Charging.
The question is, Can Mechanized Battle Armor Dismount from a Sprinting Mech?
For the sake of keeping the game moving and not finding anything specific, we ruled they could, as it is more an Infantry Action than the carrier's Action. Still, it would be nice to have an official answer.