In the original game, if you pick up a Clock in a room with Wallmasters, they will get stuck in the wall. If it happens to be a room where the doors are locked until all enemies are defeated, your only option is to reset or use the "quick end" code.
Due to the nature of keys in this game, it was possible to carry keys to other dungeons to unlock their doors. This can wind you up in a bind if you run out of keys. Future games corrected this by restricting key use to the dungeon they're acquired in.
Link's Awakening was notorious for a door that could only be unlocked with an Interchangeable Antimatter Key. The door in question was located across a moat that you were meant to cross after gaining the ability to swim. The designers failed to realize that Link could just barely clear the moat by jumping, leaving him with neither the swimming ability needed to finish the dungeon nor the key needed to get that ability. The DX version fixed this by making the jump impossible.
You can also get stuck in Eagle's Tower in the original (non-DX) version. If you drop one of the orbs down a certain hole, then it won't respawn properly, preventing you from completing the dungeon.
It's also possible to drop the orb behind a pillar and on spikes, rendering it impossible to retrieve it.
You can also trade your shovel for the boomerang, and then another shovel will appear in the shop. Buy this shovel, and then your inventory will be too full in the last dungeon to pick up the Fire Rod, which you will need badly. This can be rectified by using up all your Bombs or Magic Powder, thus freeing an inventory slot... but without Magic Powder, the final boss becomes unbeatable...
The infamous "Map Glitch" which allowed the player to skip whole screens at a time would, in theory, make the item Fetch Quest easier. However, due to a glitch in the programming, only the next item in the player's quest will be available. Skipping ahead to the final item (the magnifying glass) if you don't have the second-to-last item when you arrive there, your next item will be sitting there. You will have a very hard time winning the game from that point, since you will not be able to complete the maze at the end without either brute-forcing every combination of moves or using the very same glitch to warp directly into the final boss's room. Also, if you are unfortunate enough to save the game inside the magnifying glass cave after performing said glitch, you will never be able to leave since the mermaid statue is still blocking the entrance on the overworld.
Farting around with this glitch in caverns has a pretty good chance of dumping you into a facsimile of the sixth or seventh dungeon long before you're supposed to enter. If you try to exit, you'll wind up stuck in the dungeon's still-hidden entrance and unable to move. This and the above are all good reasons to not experiment too much with the glitch until you've learned the Manbo Mambo and can warp out of these unwinnable situations.
Thanks to a glitch in the game, players can turn any item in their inventory into a bottle. This is generally regarded as very helpful when used for useless things like the receipt for the Biggoron Sword. If it's used on something you actually need, you are SOL.
In the Wii version, there's an infamous glitch where saving and quitting at the scene where you find the big cannon before talking to an NPC makes the game unwinnable. Several known workarounds exist to allow players to talk to the invisible NPC and allow them to escape the "unwinnable" situation.
There's a game-halting bug that occurs in both versions. After crossing the Bridge of Eldin and bombing the rock wall, a piece of the bridge will be warped away to somewhere else. If you happen to save and quit between here and the Twilight, then you can be reloaded on the wrong side of the broken bridge with no way to get back across.
This can be fixed, but it requires a bit of glitching of its own. Video here.
It's possible to use the hammer to enter the Goron Temple without first becoming an honorary Goron. If you claim the pure metal without doing that, then your fairy won't let you leave until you become a Goron... which you can't do because one of the Gorons grew up. You can no longer talk to every Goron, which is one of the requirements for becoming one. But it's totally worth it because you get to see two Gongorons.
If you get the Hourglass, save right away, turn your game off, and start right again, then the hourglass is gone. Lost Forever. As you can probably imagine, you cannot complete Phantom Hourglass without the Phantom Hourglass.
The Song of the Hero quest is impossible and the save could be corrupted should the player speak to Golo the Goron a second time in Lanayru Mine, after getting the Thunder Dragon's song, but before going to the forest or volcano regions. If you save instead of loading an earlier save, prepare to restart the whole thing. Luckily, it's easy enough to avoid, and Nintendo has posted a free downloadable channel that corrects the save corruption.