RAW, frostbite has no adverse effect on heat metal for damage
There are no examples of a damage-causing spell nullifying the damage of another. It is, of course, possible for advantage/disadvantage granted by one spell to be negated by the opposite effect caused by another spell, and for non-damage effects such as darkness to be dispelled.
RAW, the target would take both fire and cold damage in your example.
Even in reality, it's possible to suffer from fire burns and ice burns simultaneously - take for example mountain climbers at very high altitude suffering both sunburn and frostbite. One type of burn doesn't negate the other.
What the spells say...
You cause numbing frost to form on one creature that you can see
within range. The target must make a Constitution saving throw. On a
failed save, the target takes 1d6 cold damage, and it has disadvantage
on the next weapon attack roll it makes before the end of its next
Heat Metal says:
Choose a manufactured metal object, such as a metal weapon or a suit
of heavy or medium metal armor, that you can see within range. You
cause the object to glow red-hot. Any creature in physical contact
with the object takes 2d8 fire damage when you cast the spell. Until
the spell ends, you can use a bonus action on each of your subsequent
turns to cause this damage again.
If a creature is holding or wearing the object and takes the damage
from it, the creature must succeed on a Constitution saving throw or
drop the object if it can. If it doesn’t drop the object, it has
disadvantage on attack rolls and ability checks until the start of
your next turn.
They say nothing about interference.