If a battlefield was not conquered at the end of the turn, do the remaining units do damage at the beginning of the next turn?

No, units do not deal damage at the beginning of the next turn.

According to the Showdown Flow and Rule 460, combat damage only occurs during the Combat Damage Step of a Showdown or Combat. Combat is a specific event triggered by units moving to a battlefield (Rule 447) or being present at a battlefield where a Showdown is ongoing.

Here is the breakdown of why this does not happen:

1. Combat is not a turn-based event: Combat damage is not a "beginning of turn" effect. It is a specific step that occurs only during a Showdown or Combat sequence (Rule 460).
2. Cleanup: At the end of a turn, the game performs an Ending Special Cleanup (Rule 317.2.a). During this process, all damage on units is healed (Rule 317.2.b), and all "this turn" effects expire (Rule 317.2.c). Any combat that was ongoing would have already concluded its damage step and cleanup before the turn ended.
3. Beginning Phase: The Beginning Phase is primarily for Holding (scoring points for controlling a battlefield you started the turn with) and other triggered abilities that specifically reference the "Beginning Phase" (Rule 383.4.d). There is no rule that causes units to deal damage to each other simply because they are present at a battlefield at the start of a turn.

If units from opposing players are present at a battlefield at the start of a turn, they are simply occupying that battlefield. They will not deal damage to each other unless a new Showdown or Combat is initiated (e.g., by moving a new unit there or using an ability that triggers combat).