I just listened to this audio recording of an interview with Tolkien. We have it straight from the man's lips!

Edit: That recording is fake. It's not really Tolkien. There's been enough doubt (and doubt whether I realize this) in the comments that I'm including a disclaimer.

Okay, Tolkien's answer aside:

The biggest (in-universe, rather than narrative) reasons everyone cites are how using the Eagles would've compromised the mission's secrecy, Sauron had the Nazgûl and archers for air defense, and the Eagles' unwillingness to be used as glorified air taxis.

I recently read two more reasons that I thought were pretty compelling:

One, since the Eagles are intelligent beings, the Ring would have tempted and corrupted them. Especially if they flew directly over Mt. Doom where its influence was strongest. Gwaihir would have tried to seize it for himself. (Landroval might have then figured he'd make a better Windlord than his brother.) Mordor's airspace is probably just as subject to Sauron's corruption as the rest of the Black Land.

Two, the plan is incredibly dangerous and has a high chance of getting Eagles killed. Yes, they're a goodly race that will fight Sauron on behalf of the other Free Peoples. But they aren't going to risk their lives unnecessarily. The Fellowship doesn't need them to get to Mordor. They helped Gandalf twice because he'd have remained imprisoned in Orthanc or (probably) died of exposure atop Zirakzigil if they hadn't. The Fellowship can manage fine without an air lift.