I think for the kind of games I run, running out of daily resources is analogous to dying, though clearly not nearly as severe. That is, I think the threat of either should be a lot more common than the actual occurrance.
The players should be
worried about running out of daily resources. To pull this off, they necessarily must occasionally so run out. But you can also get somewhere with the same tricks that make dying seem more likely than it is: Uncertainty, color, dramatic reversals, etc.
I've said before that when you play any edition of D&D as if it were a boardgame, you'll get an experience that is a lot more like a boardgame. If you use every resource (every pawn, knight, etc.) at your disposal, every time you sit down to play, then you'll move the game closer to a boardgame. If you push the characters to their limits every time (actually push them, not make the player think they are so pushed), then players will drift into trying to milk every resource.
This is true of any edition of D&D, though in my opinion, 4E and 1E (using the random dungeon generator tables) will work a bit better under such limited conditions. That is hardly an endorsement for playing that way, though, or at least not when more engaging options are open for the taking.
So that is the long way around for me to say that if running out of dailies is occuring on a regular basis, it is a symptom of another issue. In itself, it really doesn't matter that much, which is why it can happen some (and really should) with no problem whatsoever.