Strictly Rules-as-Written, this is the DM's call
As far as I can tell, there's no passage in the Player's Handbook that explicitly specifies that damage rolls for ongoing effects that trigger damage multiple times are, or are not, re-rolled for each instance of the damage. The closest it gets is when it specifies that for Area-of-Effect spells, or spells that otherwise deal damage to multiple creatures simultaneously, only a single damage roll is used for all creatures hit in a single instant (Player's Handbook, pg 196).
As DM, I have always ruled that damage effects are re-rolled for each instance of damage
I think there are a few good reasons to support this ruling.
The biggest reason, in my opinion, is that "roll once" tends to turn the damage rolls into a kind of Goblin Dice Problem, where the fate of an entire battle can hinge on a single roll. If the initial damage is high, the spell wipes the entire enemy combatants (or the party!) in only a few rounds with little-to-no recourse. If the initial damage is low, then you've effectively wasted a spell slot (or whatever resource is fuelling the effect). That might be mitigated somewhat with spellcasters who [probably] have to concentrate on their ongoing damage effects (which makes it a non-guarantee that they'll maintain the same spell for the full duration anyways), but they often have ways to protect their concentration, and it doesn't solve the problem.
Another reason I think it's probably not good to use the same roll over and over is that most damage sources in the game are always rolled at the moment they deal the damage. It's inconsistent to have one (or a few) type of effect that instead rolls once and is applied multiple times, and having these exceptions carved out tends to result in players being unsure what they're supposed to do when asked how much damage a creature takes. It's far more acceptable if the spell itself is written that way, or if a Class Feature is overriding these rules.
The last reason I'll offer to support re-rolling the dice for each instance of damage is that it is more useful for role-playing/storytelling purposes. If, for example, the damage from a spell like Heat Metal happens to be high on a given turn, the DM could narrate how the monster tries to rip their armor off and scalds their hand in the process. If the damage is low, maybe there was a convenient bucket of water that the monster used to temporarily reduce the pain. I think this is probably more satisfying than having to narrate how the spell was botched and won't deal much damage.