Gandalf and Those Damn Eagles
We all know that there's a huge plot hole in The Lord of the Rings: Why didn't Gandalf just ﬂy an Eagle to Mordor and destroy the One Ring? HISHE did one of their ﬁrst videos about it, and there's an elaborate fan theory speculating that Gandalf might actually have been planning to do exactly that.
Well, why didn't he? It turns out there are several good reasons.
The dull, out-of-universe explanation is, obviously, that it'd make for a boring story, and Tolkien wanted to tell a greater epic. It's doubtful that Tolkien even thought of the eagle plan, though. He doesn't address it in any of his letters, and if he was aware of the plot hole while writing The Fellowship of the Ring, he'd certainly have somebody suggest it at the Council of Elrond, only to have the idea officially shot down by Gandalf or Elrond.
But, as it turns out, it'd probably be a bad idea, even in-universe.
The eagles are not subject to Gandalf's command. As Maiar of Manwë, they are in fact his equals. In The Hobbit, it's made very clear that they help him out in that story not because he orders them to, but because they owe him a favour: He once saved their leader, Gwaihir the Windlord, from a poisoned arrow. It's entirely possible that they would refuse. However, given that they're on the same side as Gandalf (and enthusiastically join the ﬁght against Sauron's forces later on), they'd probably only do so if they had good reasons to. However…
With the sole exception of Tom Bombadil, the One Ring's corrupting influence on creatures appears to be proportional to how powerful they are. Gandalf doesn't even dare touch it. The one time he handles it (in Fellowship), he does so using tongs. Likewise, Galadriel avoids touching it, and even its presence is enough to create a temptation she has to actively resist. This is precisely why Gandalf is confident in entrusting it to a hobbit — hobbits are generally humble beings without any major magical abilities or lust for power; they typically don't even want influence on the greater events of the world. They're not immune (as Sméagol's sad story attests), but their nature makes them less vulnerable to its corrupting influence than humans, and especially Maiar and high elven royalty. The eagles, also being Maiar, would almost certainly be corrupted and turn to evil if they were given the ring. They'd certainly be aware of this (having been actively ﬁghting Morgoth's agents since the First Age), and so would Gandalf. Thus, he probably wouldn't even ask them — and if he did, they'd probably refuse.
Of course, they might be able to withstand its influence if they were merely tasked with carrying a hobbit rider who carried the ring, rather than carrying it themselves.
Mordor's Air Defenses
We know, from The Hobbit, that the Eagles avoid going near Laketown (and human territory in general) because they're afraid of archers. They've been shot at in the past while attempting to raid livestock, and this in fact nearly claimed the life of Gwaihir, their leader. The archers of Laketown are mostly shepherds and ﬁshermen with simple bows, whereas Sauron has thousands of highly-trained orc, Easterling and Haradrim military archers stationed in Mordor. Furthermore, Sauron has his own air force, in the form of the fell beasts, notably used as mounts by the Nazgûl. It's entirely likely that an eagle could defeat a fell beast in a ﬁght (in the movie adaptation of Return of the King, we actually see an eagle defeating a beast-mounted Nazgûl), but a single eagle would be hard pressed to handle a group of them. This would necessitate sending an entire group of eagles, and that would effectively preclude a stealth mission. Without the element of stealth (and Sauron would be able to spot a squadron of giant eagles from half a country away), they'd be forced to deal with the entire massed forces of Sauron's archers and however many fell beasts are at his commandThere's an entire region of Mordor named simply "The Vale of the Fell Beasts", so it seems likely that there were more of them than the Nazgûl's mounts. We know, for a fact, that they're both afraid of archers, and sufficiently vulnerable to archery that they've been harmed by archers in the past.
On the other hand, we also know that the Eagles can ﬂy at high enough altitudes that even Aragorn's superhuman eyesight isn't able to clearly see Gwaihir ﬂying overhead in The Two Towers. An English longbow has a maximum range of about 350 meters, and the bows used by orcs, Easterlings and Haradrim would probably be considerably shorter-ranged than that — particularly those of the Haradrim and the Easterlings (noted as by far the deadlier of Sauron's archers), who used short compound bows, and the latter of whom even favoured horse archery. So if the eagles could just stay at a very high altitude, then they should be safe from archers (and only have to deal with fell beast interceptors). Well…
Approaching The Cracks of Doom
That HISHE video aside, throwing the ring into the caldera of Mt. Doom explicitly isn't enough to destroy it, it has to be thrown into the Cracks of Doom, the underground workshop where the ring was forged. The only entry to the Cracks of Doom is a long narrow tunnel bored into the mountainside. In other words, an eagle couldn't simply drop the ring from high above, it'd have to land and set a rider into the tunnel. This would require it to give up the aforementioned altitude advantage, and it'd be vulnerable to any archers Sauron had stationed at the mountain.
In neither book nor ﬁlm adaptation did we see any archers on the mountain, and there likely weren't any. That's because Sauron wasn't aware that anybody was trying to get to Mt. Doom — he was aware that two hobbits had infiltrated Mordor, but (given what the Mouth of Sauron says during the parlay before the Battle of the Black Gate), he apparently believed them to be on an espionage mission. If, instead, he had spotted a squadron of giant eagle Maiar on a course for Mt. Doom, he'd have an entirely different response than he did to three lesser humanoids attempting to sneak into his lands.
When the eagles did enter Mordor and ﬂy to Mt. Doom (on their ﬁnal rescue mission to fetch Frodo and Sam), they did so only after Sauron himself and all of his armies had been defeated, after the destruction of the One Ring.
Even if Gandalf had asked the Eagles to ﬂy to Mordor and destroy the Ring, they'd probably have refused, and they'd be right to.