Does damage dealt to a commander count toward its owner's life total in a Commander game?

Damage dealt to a creature — including a commander — is marked on that creature and reduces its effective toughness for the turn. It does not directly affect any player's life total. This is standard damage rules per CR 120.1 and CR 510.1.

What you may be thinking of is the Commander damage rule, which works in the opposite direction: damage dealt by a commander to a player counts toward that player's life total loss, and if a single commander has dealt 21 or more combat damage to a player over the course of the game, that player loses the game (Commander rule, CR 903.10).

The only way damage to a commander affects life totals is indirectly through lifelink (CR 702.15): if a creature with lifelink deals damage to a commander, the lifelink creature's controller gains that much life. Conversely, if the commander itself has lifelink and deals damage, its controller gains life.

Concrete example: Your opponent swings with a 5/5 and deals 5 damage to your Ur-Dragon commander (a 10/10). Your Ur-Dragon now has 5 damage marked on it and will die if it takes 5 more before end of turn — but your life total is completely unaffected by that 5 damage.

In summary: damage to a commander affects only the commander's survival (state-based actions per CR 704.5g), never the owner's life total.