This article is from the Tolkien Newsgroups FAQ, by Steuard Jensen firstname.lastname@example.org with numerous contributions by others.
This debate is most interesting when limited to "story internal"
arguments (i.e. "Otherwise it would be a dull book" is a cop-out). One
possibility is that there were no Eagles available when they were
needed. Another common argument is that Mordor was well defended,
while the Eagles were wary even of shepherds with bows. It may have
been an issue of secrecy: a group of Eagles far from the Misty
Mountains flying toward Mordor might well have drawn Sauron's
attention, and might have even hinted at the plan to destroy the Ring.
Some suggest that like the Istari, the Eagles were forbidden by the
Valar to help so directly. Another suggestion is that Frodo may have
needed the long journey to (hopefully) develop enough strength of will
to cast the Ring into the Fire. Many other explanations are seen
repeatedly as well. In the end, most participants tend to agree that
an Eagle taking the Ring to Mount Doom would not have worked.
But that is not the end of the discussion. Even if those objections
are valid, many believe that this topic should have come up at the
Council of Elrond (after all, the book shows discussions of other
flawed options there, like sending the Ring to Bombadil or throwing it
into the Sea). And some people still think that making use of the
Eagles would have been effective (even if only for part of the journey,
rather than flying all the way to Mordor). These aspects of the
question remain unresolved.