Does damage marked on a creature carry over between combat and the next phase?

Damage marked on a creature persists only until the cleanup step at the end of the turn (CR 514.2). At that point, all damage marked on all permanents is removed simultaneously. This means damage does not carry over from one phase to another between turns, but it does persist within the same turn across phases.

For example, if a creature is dealt 2 damage during the combat phase but survives (say it has 4 toughness), that 2 damage remains marked on it through the postcombat main phase, the end step, and is only wiped away during the cleanup step. It does not persist into the next turn's phases.

A creature dies from damage only if it has damage marked on it equal to or greater than its toughness at a state-based action check (CR 704.5g). State-based actions are checked continuously, so a creature with lethal damage will be put into the graveyard before the next phase even begins — the damage never needs to 'carry over' for lethal purposes.

Example: Your 2/2 creature blocks a 1/1 and receives 1 damage. Both survive combat. During your postcombat main phase, an opponent's spell deals 1 more damage to your 2/2. State-based actions now see 2 damage on a 2-toughness creature and it dies — the combat damage carried over within the same turn and combined with the new damage to be lethal.

The key rule to remember: damage is removed in the cleanup step (CR 514.2), not between phases. Within a single turn, marked damage accumulates across all phases and steps until cleanup.