The "you're low on hearts" alarm appears in every single game, whenever Link is damaged too much: BEEP. BEEP. BEEP. BEEP. BEEP. BEEP. BEEP. If you ever played the first or third LOZ games, these sounds will haunt you for the rest of your life. Even if it has been ten years since you played either. You will never unhear it.
Downplayed in some later games. It's still there, but they made the sound softer, quieter, and less frequent.
You'll quickly tire from hearing Link getting hit and the Game Over jingles, especially when it's a difficult dungeon.
The recorder song, simply because of its high-pitched notes on the NES version. It will quickly get on your nerves by the time you leave Levels 7 in the first quest and Level 8 in the second quest because of the Degraded Boss, Digdogger, who is only weak to that. Downplayed in BS The Legend of Zelda, which makes the song's instrumentation less high pitched and annoying.
The sound of the Wizzrobes firing their magic (BWO-O-O-OOOO!) isn't too annoying by itself, but when you have to fight several of them at the same time (which is often), it becomes a nightmare; not just because you'll hear it a lot, but because these guys are incredibly strong, capable of killing Link in roughly 8-12 hits with their magic, and as little as 2-4 with Collision Damage (particularly with the red ones). And if you do get killed, well...Hope you have earplugs! The blue ones in particular can really get on the player's nerves because they spam the attack whenever they are walking in Link's direction.
In the Game Boy Advance port of The Legend of Zelda: A Link to the Past, Link now Kiais each time he swings his sword. Ocarina of Time and Wind Waker had the excuse of giving Link a somewhat varied "vocabulary", him being a mute hero and all, but this version... not so much, and ALttP has lots of enemies that require multiple slashes in rapid succession to defeat, resulting in a cacophony of "HAH!HOH!HWAH!AAH!HWAH!HOH!HAH!!" repeated ad nauseam.
Any game with the ReDeads. Their scream is not only annoying, it means you're paralyzed for a few seconds while they creep towards you to kill you, making them an unholy mix of this trope plus Hell Is That Noise.
Though the Guardian Acorn/Pieces of Power from The Legend of Zelda: Link's Awakening seem like useful items at first (they double your defense/attack power until you take a certain number of hits) the entire time they're in effect they completely replace the background music with an annoying, repetitive drone that does not go away unless you enter a building or take three hits from an enemy so the effect can wear off. There are certain tunes that will override the powerup theme, though, most notably the boss battle themes.
This is no longer the issue in the Switch remake, as both of the items were nerfed to only last for a limited time, regardless of how many times Link has been hit.
Zelda's Lullaby. If you weren't sick of playing it by the time you got to the Water Temple, you most certainly were after that, due to the fact that it's used to change the temple's water levels.
The Owl's theme music, mostly by association.
"HEY! LISTEN!"
Zelda's shrieking will quickly get on your nerves on the final boss fight because you hear it every time you get hit.
When you are in the Goron Shrine, the horrible grating noise that the crying Goron child makes will drive you bat-crazy in a short amount of time. Then again, the other Gorons in the city are just as sick of it as you are. (Fortunately, the ocarina song for that region is a lullaby, which even works on all Gorons.)
Tatl's call is a bicycle bell. Though, they did change it based on complaints from Navi's "Hey! Listen!", and to make it less frequent, so at least it's a step up.
The Skull Kid's laugh.
The Elegy of Emptiness. By the time you get to the top of the Stone Tower, it's gotten old. And then, you have to go inside. Though the sound itself isn't the problem. This one is longer and more complex than the simple "hit up-down-up twice" idea with most of the songs. It requires the most work and then you have to wait for it to repeat and for stuff to happen. Most of the puzzles in Stone Tower require playing this song six or seven times. It is incredibly, insanely long, boring, and tedious and it makes things like OoT's Water Temple (the returning to the equipment screen to switch boots every five seconds, the former standard for rage-inducing tedium in a Zelda game) seem fun by comparison. You will go Ax-Crazy at the sound of it by the end.
Odolwa's constant chanting/shouting.
From both Majora's Mask and Ocarina of Time, the howling of Wolfos. It's annoying even the first time you hear it.
The song to change the direction of the wind gets very old very fast. Fortunately, you'll hear it less often in the HD version if you buy the Swift Sail after it becomes available, which automatically changes the wind direction (though you'll still need the song for purposes other than sailing).
The Command Melody gets old faster, despite learning it comparatively late in the game.
There are enemies called Miniblins that, although small and weak, have a tendency to appear in (seemingly never-ending) swarms. They make a high-pitched noise like "da-NA! da-NA! da-NA!" AND THEY WON'T SHUT UP until every single one of those annoying, respawning things is DEAD.
Fighting a Wizzrobe and hearing another one spawn and summon more monsters gives a solid Oh, Crap! feeling.
Kargaroks make this "EYAAA" sound...over and over and over and over until you kill them simply because you don't want to hear the sound...then another one shows up.
The strange not-quite voice acting, although tolerable, got to critical levels when trying to repeatedly buy things from Beedle: "THEEENNK YEW!! THEEENNK YEW!!"
Pressing the A button on the Game Boy Advance when using the Tingle Tuner causes Tingle to scream "Heeeeeey!" while the screen zooms in on Tingle's cursor.
The repetitive and unskippable cutscene and music note every time Link picks up an item and holds it up or opens a chest and almost falls inside it.
The horn Bokoblin leaders use in to summon other Bokoblins. DEAR GOD. And it doesn't stop until you kill the one that's using it. Thankfully, after you get the whip from the Ancient Cistern, you can just take the horn from the Bokoblins.
Whenever you use your sword for dowsing. BOOP. BOOP. BOOP. BOOP. BEEP BEEP BEEP BEEP BEEP.
"HEY LISTEN" combined with the heart alarm — when your hearts get low for the first time in a given gameplay session, Fi will want to tell you that your hearts are low and you should get some health. Sure, you could just ignore her, but her hails add another chime to the already annoying alarm.
The Wizzrobes' vocal cords appear to be developed to sound as dissonant and annoying as possible. Especially when laughing, which they do every time they stun Link with their lightning bolts or ice, while dancing safely out of reach of Link's sword. HE-HE-HIEEUW! HE-HE-HIEEUW!
The weird squealing noise that Kilton makes. It wouldn't be so bad if he didn't do it every time you buy or sell something in his shop, so if you're planning on unloading a lot of monster parts all at once, you're going to hear it a lot.
Skull Kid uses his ocarina to order his fairies to attack as his basic combo. Unfortunately, keeping with his portrayal from the beginning of The Legend of Zelda: Majora's Mask, he only knows one note. Thus, the note gets played incessantly while using him.
Tetra's repeated shout during her basic attack chain is often compared to a duck by players.
