Both the random 'Perpetual Questing Initiative' quests and the changing 'Dungeon That Happens To Be On Fire And Also This Is Somehow A Good Thing' appear to serve the same purpose - to give postgame players some further direction (and income) for their adventures. It's nice to have options to choose from if you don't overly feel like trudging through dungeon X at the moment, but when compared side by side the two systems feel a little unbalanced.
The PQI awards you 500 extra gold for winning a specific dungeon, using a specific race / class, and achieving a specific badge. This can often result in quite tricky combinations, especially when you consider that the quest will most likely ask you for a class or badge you haven't won with yet because it's difficult (or you're poor with it) in that dungeon.
The burning dungeon, meanwhile, allows you to use or repeat any class or tactics you want without a target challenge. Yet it still awards you 450 extra gold* for what is in practice a significantly easier task. Unless the PQI happened to overlap with some other unfinished quest, or indeed the burning dungeon itself, I can't imagine it feeling worth the trouble of checking with the rewards for each as they currently are.
Perhaps the gold rewards could be adjusted further apart?
* Compared to a nonburning dungeon both prepped with 'Bet on Boss', because unless you explicitly need the purist badge for the first time why wouldn't you ... which honestly feels like its own obvious-choice problem most of the time.