I think it would be even more dramatic if Link...
finished off a boss, gave the beast its death-blow, but then a cut scene happens where you see the creature's death-throes and, in them, it shoots a bolt of magic toward Link or whips out a tenticle or a tail that grabs him and lays a poison sting into him. The the creature dies, explodes, and... get this... you DON'T get your heart container right then! (You wind up getting it after you complete the heal quest). Your screen or your partner (if you have one in the game) informs you that you've been poisoned and need to get help quick! Link starts his life-run spry enough, but then he starts to stagger, the screen blurs in and out, maybe when he's almost done for he start "uffing" and yelpng like he does when something is hurting him... doubles over and pants when the heart gauge gets low like he does - but with the added in-out focus of the screen and the staggers...
And, when you reach the healer, you get a little cut screen, Link wakes up in a bed and sits up, the healer tells him that he was almost a goner, scolds him on being so reckless with himself, and presents him with "something to keep you healthy in the future" - the Heart Container!
Actually picking it up should be optional, for all those Zelda fanatics who like to do the three-heart challenge.