Would it be overly unbalanced if a DM were to houserule that two or more instances of resistance to the same element (such as fire) to be treated as immunity?
It allows for some explosive combat options and methods to gain extreme immunities
For example, a Tiefling gains resistance to fire damage, they could walk into a group of enemies and cast fireball on the ground and then use absorb elements to gain immunity to the damage of the spell and also all fire damage until the start of their next turn.
There are also certain combinations that effectively make a character completely immune to damage. For example, a Bear Totem Barbarian at level 3:
While raging, you have resistance to all damage except psychic damage. The spirit of the bear makes you tough enough to stand up to any punishment.
They could have the warding bond spell cast on them which states:
While the target is within 60 feet of you, it gains a +1 bonus to AC and saving throws, and it has resistance to all damage.
This would thus grant them immunity to all damage except psychic for a minute. Alternatively, even without the warding bond they would have resistance to all but psychic and then their rage would also grant them resistance to bludgeoning, piercing, and slashing which would take into immunity to those three damage types.
Another problematic combination is a 14th level Abjuration Wizard with Spell Resistance:
[...] Furthermore, you have resistance against the damage of spells.
And then also a 7th level Oath of Ancients Paladin as they have an Aura of Warding:
[...] You and friendly creatures within 10 feet of you have resistance to damage from spells.
This would grant the Abjuration Wizard immunity to the damage from any spell.
It also may make abilities which allow you to ignore a creature's resistance to a damage type less helpful (it would be GM dependant how this rules and features of that kind interact).
Yes. It is a game-breaking house rule.
Stacking traits are always problematic. The design of 5e allows for stacking traits only sparingly, unless we suffer the same kind of Cleric of Doom problem we had in 3.5e.
Some scholars of the game meta say that every encounter is designed to drain X resources from the party. Spell slots, HP, equipment, potions, long-rest abilities, short-rest abilities (like the fighter's Action Surge).
Immunity to damage breaks that premise. It hurts the game balance. Besides, if we think of resistance as a halving of damage (including the rounding down), one could argue that two resistances would cause a quarter of the damage to go through. And so on.
That, however is not fun and not simple. Creating such combos is neither fun, nor fair in the game. Especially when one can, at the 3rd level plus one magic item, become invulnerable with the help of a friendly Cleric.
Barbarian 3 (Bear totem): [...] While raging, you have resistance to all damage except psychic damage. [...]
Ring of Psychic Resistance (requires attunement): You have resistance to psychic damage while wearing this ring.
2nd level cleric spell: Warding Bond: While the target is within 60 feet of you, it gains a +1 bonus to AC and saving throws, and it has resistance to all damage.
P.S. It seems a cleric cannot use Warding Bond on themselves.
Note that WB does not require concentration so you can rage. As long as you make an attack and does not end rage, this character is immune to any damage if that house rule is in effect.
So, yes. It is a game-breaking house rule.
In 1e, we had such immunity, but nearly only to elements such as fire, water, electricity... Not to all types of resistance. Also, races in 1e did not have any kind of resistance in comparison to the Tieflings or Dragonborn in 5e. Well... Elves were resistant to "charm" type of spells. But most resistance powers were more bonuses to saving throws which was not giving you that much.
I think that the 5e rules of disallowing cumulative resistance in most cases is for that simple reason: it would make the characters and monsters way too powerful. To a minimum, you could accept accumulation by dividing the damage by a power of 2. So one instance of resistance to the damage type causes the creature to take 1/2 damage, two resistances cause 1/4 damage, three resistances cause 1/8 damage, etc. There are already some cases where 5e allows 1/4th the damage (on a successful save when you have resistance).