Removing mana burn opens two design spaces: Life Total and Mana Pool. Think about how much better Threshold, Madness, Flashback or Hellbent would be if there was some random ability called "Card Burn" which allowed you to just discard cards at the end of any phase. In Alpha and Beta when the rules were being made up, Card-Burn would not have been a problem. But that means these graveyard mechanics would not have existed. Card-burn ultimately would have been a design-space closing drawback.
Here are a few cards that I made up for this post. I'll intentionally leave the cost blank. Think about what this card would cost both with and without manaburn...
No Pain No Gain

-Enchantment
At the end of each turn, if you lost life this turn draw a card.
Empathy

-Enchantment
If damage would be delt to you by another player, prevent 1 of that damage if your life total is within 3 life of thier life total.
Cannopy Cover

-Enchant
Creatures controled by players with more than twice your life cannot attack you.
Noble Research

-Instant
Draw a card.
If you have less than 10 life draw an extra card.
If you have less than 5 life draw an extra card.
Thaumaturgist

-Creature
0/1
Whenever you lose life, if there are less than four counters on Thaumaturgist put a +1/+1 counter on Thaumaturgist.
Remove two +1/+1 counter: add

or

to your mana pool.
There are also a ton of interesting mana pool cards you could use as well. These cards would likely be terrible if your mana pool delt you damage:
Eyes of Dominance

-Enchantment
Creatures with less than the amount of mana in your mana pool cannot attack you.
Eyes of Supression

?-Instant
If you have more mana in your mana pool that target player, that players emptys mana pool.
Eyes of Betrayal

?-Enchant Aura
At the begining of enchanted creature's controler's upkeep, the player with the most mana in thier mana pool gains control of that creature.
Eyes of Doubt

-Instant
Counter target spell if that spells controler does not have 3 or more mana in there mana pool.
Eyes of Passion

-Instant
Target creture gets +X/+X where X is the amount of mana in your mana pool. And -Y/-Y where Y is the amount of mana in all opponents' mana pools.
Really the ONLY card that I can think of that has absolutly no application in a world without mana burn is Power Surge simply because it occurs "before the untap step." Citadel and Port and whatnot still primarily serve to tap out a player durring your turn. If anything those cards are now -Stronger- against bad players because now a bad control player may opt to foolishly tap out to avoid damage rather than leave open mana to control you.