Does commander damage from a commander dealt during a previous game carry over in a best-of-three match?

Each game in a match is a completely separate game. When a new game begins, all game state is reset: life totals return to their starting values, cards return to their owners' decks (or command zones), and all counters, damage, and tracked values from the previous game are wiped. Commander damage is a game-state value tracked during a single game, not across the match.

The Comprehensive Rules state in CR 103.1 that at the start of a game, all players begin with a clean game state. Commander damage (the rule that a player loses if they have been dealt 21 or more combat damage by a single commander) is governed by the EDH rules and CR 704.6c, which checks this total only within the current game.

There is no rule — in the Comprehensive Rules or the official Commander format rules — that carries any damage totals, counters, or life total changes from one game to the next in a match. Each game stands entirely on its own.

Concrete example: In Game 1 of a best-of-three Commander match, your opponent's Zur the Enchanter deals you 18 combat damage before you win. In Game 2, Zur's commander damage counter for you starts back at 0 — your opponent must deal a fresh 21 combat damage to eliminate you via that win condition.