IN the War of the Ring, Thranduil's forces were busy fighting in Mirkwood--with the "Woodmen" or Beornings no doubt against the forces of Dol Guldur. I suppose that they really couldn't come galloping to the defense of Gondor any more than could the Dwarves or Erebor or the men of Dale: they were too busy fighting in or around their own lands.
The Rivendell(and presumably Linden) elves seem to be more inactive--they're weary of Middle-Earth and many of them are just biding their time to leave. But they do play an active role in supporting the Dunedain in their defense of the few habitable places left in Eriador, not to mention their fostering of Aragorn, Valandil and possibly others.
Earlier in the Third Age they were quite active against Angmar and they did fight alongside men of both Gondor and Arthedain(and what's left of Cardolan?) against the Witchking. It is Cirdan's Elves who search for and find Arvedui amount the Ice-men of Forochel. And forces from Rivendell and Lindon fight with the armies from Arthedain and Gondor.
Perhaps by the time of the third age they were too few and too weary to do much else? Or perhaps they too were busy defending their realms from the various beasts and "ruffians" that were marauding around Eriador?
This is just the Third Age. Of course in the Second Age there was the Last Alliance, and the entire Silmarillion is an account of the Elves of Beleriand fighting a centuries-long war against Morgoth.
Appendices A and B in the LOTR is the source of a lot of this information--there are probably others as well.
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