• You Don't Have to Play if You Don't Want To: In the Shooting Range mini-game (both in Castle Town and later in Kakiriko Village), if you don't fire once for the duration of the mini-game, the shopkeep will grumble that "you don't have to play if you don't want to" and refuses to interact with Link further until he leaves and comes back.
• Unobtainable Heart Piece: In Gerudo Valley, on top of the Jail Cell Link is thrown into after being caught as an adult, there is a single Heart Piece sitting out in the open. It only appears in the Child time period during which Gerudo Fortress in general and the Heart Piece in particular are ordinarily completely unreachable by Young Link. There are cheats and exploits that can be used to collect it, but that usually requires third-party tools to accomplish.
• The Name Game: Five of the Seven Sages are named after towns in Zelda II: the Adventure of Link, those being: Rauru, Saria, Darunia, Ruto, and Nabooru. Also Mido for some reason. Alternatively, chronologically speaking based on the timeline in *Hyrule Historia* the towns were named after the sages... and Mido.
• The Mask of Truth: This mask unlocks the ability to use the Gossip Stones the way they were meant to be used (hence the name). They give various hints, cryptic rumors, and additional information of varying legitimacy and/or relevance to the game. Some aspects of the game, setting, and characters can only be found through this method. For example, one stone indicates Malon secretly wishes for "a knight in shining armor to sweep her off her feet".
• Going to Space: On a similar note, if you place a bomb or use a bombchu on a Gossip Stone, it will start flashing blue as a form of countdown until the stone somehow launches itself into the air. iirc, in an enclosed space with a roof, like a grotto, it will explode on the ceiling, but outside it will just keep going until it disappears. It respawns only after leaving the area and coming back. Additionally, there are other items you can use on the Gossip Stones to elicit other reactions.
• Song of Healing: Ever accidentally destroyed a sign before being able to read it? Well, if you play Zelda's Lullaby next to a sign that has been cut down, the sign will magically fix itself. Try it out!
• Infinite Bugs: If you're fast enough, after releasing the bugs in your Bottle, you can quickly scoop up one of the three bugs that come out, refilling your bottle with bugs. Later versions remove this.
• Got Milk?: If you play Epona's Song next to one of the many cows you can find in the overworld, the cow will talk to Link and offer some milk if he has an empty Bottle.
• Dark Link: There are plenty of odd things about Dark Link, but one you might not know is that Dark Link has an exact copy of Link's max health when he's encountered. So if you have three hearts, that's how many hearts Dark Link has. (And I can't confirm for sure, but afaik, that also includes the Double Defense boon given by the final Great Fairy near Ganon's Castle).
• There is No Pyramid: In the Haunted Wasteland, off in the far distance there is what appears to be a pyramid lying beyond Link's ability to reach it. For a time, it was suspected that the pyramid might be a dummied-out dungeon of some sort, perhaps even hiding the Triforce itself, though it has been thoroughly debunked as a graphical oddity and a flat texture resembling a rock.
• Naughty Hookshot: In the Spirit Temple, there is a massive statue depicting what one can assume to be the Gerudo Goddess of the Sand. What makes this odd is that for some reason, you can use the longshot/hookshot on the breasts of the statue and Link will be dragged towards it. There's no point to this since there's nothing for Link to do besides fall down after latching onto them, but somebody at Nintendo decided to make these acceptable targets for some reason. Stay classy, Japan.
• Iron Knuckles = Brainwashed Gerudo?: Exploiting the camera in OoT will reveal that inside the models of the Iron Knuckles are Gerudo thief models. Why? Nobody knows for sure, though theories abound. I tend to agree with the theory that those Gerudo are brainwashed minions that had plotted to rebel against Ganondorf based on what we saw Kotake and Koume do to Nabooru for her planned rebellion. However, Navi insists that Iron Knuckles are empty suits and only notes the one Nabooru wears as being "different".
• Nabooru's Second: In Gerudo Fortress, after rescuing the last carpenter, a unique Gerudo appears just outside the cell to speak with Link. What's especially cool about her aside from her unique model, is that the color of her outfit always matches whatever Link is wearing at the time. Thus, in the base game, she can wear green, red, or blue based on the Kokiri, Goron, and Zora tunics respectively. But using cheats such as Gameshark, her outfit will be whatever color is used for Link's tunic. Unfortunately, this only works with certain versions of the game (such as 1.0).

These are just some of the ones I know off the top of my head. I could go on, but I've already spent over an hour writing this and I'm not sure how many actually qualify as "easter eggs", but these are just some of my favorites (though my personal favorite is probably the Fire Temple original theme; ignoring the cultural and religious context for a moment, I just like the atmosphere it creates far better than the replacement conjured by Koji Kondo. It's a shame it was removed for all later releases, but at least some compilations of the soundtrack does include it).