Dale is mentioned in the The Return of the King, in one of the appendices. It's a pretty short mention, but basically, a portion of Sauron's armies attacked Dale during the War of the Ring, and forced the men into the mountain to hole up with the dwarves. There was even a Battle of Dale, where both of the kings were killed (Brand and Dain). The siege lasted until Sauron's much larger southern army was defeated at Pelennor Fields, at which point the splinter group gave up and left.
Strategically, this siege was very important: it kept Sauron's army split into at least two parts, and also kept them away from the elven lands further along.
As for why it wasn't brought up in the film: it was a siege, and those are generally boring. On top of which, none of the characters in Dale had appeared in the movie up to that point, so it would have been hard to justify showing them sitting in a mountain surrounded by orcs for a week.