The Dwarves and Men of Dale did not assist Gondor because, like most other kingdoms in Middle Earth, they were facing danger and battles of their own.
At the same time as the great armies besieged Minas Tirith a host of the allies of Sauron that had long threatened the borders of King Brand crossed the River Carnen, and Brand was driven back to Dale. There he had the aid of the Dwarves of Erebor; and there was a great battle at the Mountain's feet. It lasted three days, but in the end both King Brand and King Dáin Ironfoot were slain, and the Easterlings had the victory. But they could not take the Gate, and many, both Dwarves and Men, took refuge in Erebor, and there withstood a siege.
When news came of the great victories in the South, then Sauron's northern army was filled with dismay; and the besieged came forth and routed them, and the remnant fled into the East and troubled Dale no more. Then Bard II, Brand's son, became King in Dale, and Thorin III Stonehelm, Dáin's son, became King under the Mountain. They sent their ambassadors to the crowning of King Elessar; and their realms remained ever after, as long as they lasted, in friendship with Gondor; and they were under the crown and protection of the King of the West.
From the Lord of the Rings appendices.
This attack was foreseen:
We discover that messengers have come also to King Brand in Dale, and that he is afraid. We fear that he may yield. Already war is gathering on his eastern borders. If we make no answer, the Enemy may move Men of his rule to assail King Brand, and Dáin also ’
(Gloin, in Council of Elrond)
Aside from that, they are not mentioned because Dale and Erebor are not on the way to Mordor and are uninvolved in the Quest. In the book, Sauron does also send emissaries to Erebor to seek for Bilbo and the Ring.
‘Then about a year ago a messenger came to Dáin, but not from Moria - from Mordor: a horseman in the night, who called Dáin to his gate. The Lord Sauron the Great, so he said, wished for our friendship. Rings he would give for it, such as he gave of old. And he asked urgently concerning hobbits, of what kind they were, and where they dwelt. “For Sauron knows,” said he, “that one of these was known to you on a time.”