It is technically ambiguous but precedent, Crawford, and the lack of being told to record the number all point towards rerolling the damage
This answer will borrow heavily from the question "Synaptic Static - when to roll the d6?" and the answers that already exist there.
As laid out in the most upvoted answer there by user @Ryan Thompson there is an established pattern in the game that dice rolls are used once and forgotten, there are incredibly few times where you are meant to remember the result of a die for an extended period of time. One such feature is the Wizard's Portent feature:
When you finish a long rest, roll two d20s and record the numbers rolled. You can replace any attack roll, saving throw, or ability check made by you or a creature that you can see with one of these foretelling rolls.
This feature explicitly tells us to record the numbers rolled and thus it seems likely that unless a feature tells us to record a roll, we do not record a roll. The heat metal spell does not tell us to record the roll and so we would not record it, thus the 2d8 are rerolled each time the damage is dealt.
This is also supported by lead game designer Jeremy Crawford in his tweet:
@Quadhund: Heat Metal specifically says "on each of your subsequent turns to cause this damage again." It is unclear if this damage is intended to be 2d8 or what the initial 2d8 damage was.
@Armando_doval: Ah, I see now. My money's on rolling the dice again, but let's see what Jeremy says.
@JeremyECrawford: Yes, roll again.
Here we see him stating that we should reroll the damage for heat metal, though this is also supported by the general trend of dice not having memory (not needing to be remembered) unless explicitly told otherwise.