Notable that Gauntlet was originally an arcade game and that design feature was to encourage the continuous feeding of coins into the machine to keep your playthrough going, as I recall.

I generally prefer a more exploratory, contemplative approach so it's not my favorite game feature, but as long as it's balanced and not just headless chickening, I'm not inherently opposed to it.

I do agree it doesn't marry well with randomization of maps/resources/drops.

Not quite the same, but doesn't the player character's max health in Superbrothers: Sword & Sworcery EP decrease as you progress?

I happened to remember one example that would qualify for this topic:

Shovel Knight: Specter of Torment's New Game + mode.

This mode actually makes many interesting changes to the game mechanics (though there is one change I don't like). In particular:
* Your health and magic meters are combined into one (called Darkness/Will).
* This meter drops over time, but this will not kill you.
* Getting hit when your resource is low enough *will* kill you, so be careful if you use a lot of special attacks.
* Using special attacks will drain your resource, but not as much as getting hit will.
* Anything that would normally refill your magic will refill this resource.
* The healing ability is disabled on this mode. (This is the change I dislike; I would have rather had an interesting healing ability that wouldn't have been made meaningless by this mode, like the one Plague Knight got.)

I can remember of couple of games where such mechanic existed, but not all the time.

In Pathologic there was a plague in the town you are in. And if you got infected, you would have to find medicine which would slow the progress of the desease. But it still contiued to affect you, progressing and reducing your health. So finding medicine (as well as food and water) was imperative to survive towards the end of the game.

However, I think, there was a way to avoid to be infected. Thoush it was very hard, as any contact with sick people, rats, or even some areas of the town would make you infected.

In most Sonic the Hedgehog games you fight final boss as Super Sonic. In Super form you have a ring counter constantly reducing and if it reached zero (during boss fight) you will die. So you have to collect more rings as well as beat the boss.

Death's gambit has damage per second if you get hurt by specific enemies.