Suppose that the mages Mary and Merry both cast Heat Metal at 2nd level on the metal armor of an enemy fighter, Fred. Both mages use their bonus action on subsequent turns to continue dealing damage to Fred. How much damage does Fred take each round, and when does he take it? What additional effects does Fred suffer, and how long do they last? For the sake of argument, let's say that the initiative order each round is Mary, Merry, then Fred.
Only one takes effect at any time
The spellcasting rules address this type of scenario in the "Combining Magical Effects" section:
The effects of different spells add together while the durations of those spells overlap. The effects of the same spell cast multiple times don't combine, however. Instead, the most potent effect - such as the highest bonus - from those castings applies while their durations overlap, or the most recent effect applies if the castings are equally potent and their durations overlap.
Different game features can affect a target at the same time. But when two or more game features have the same name, only the effects of one of them—the most potent one—apply while the durations of the effects overlap.
In this specific case, only one casting of heat metal will have any effect. This is because the duration of the one Mary casts overlaps with the duration of the one Merry casts (they both have a duration of 1 minute).
Which one is the "most potent" will depend on your GM's interpretation. One could use such measures as spell slot used or spellcasting ability modifier. If the GM determines that they are both equally potent, the most recent casting's effect applies.