I used the original tires that came with my first road bike on the trainer until the casing started separating because of the heat. After running out of old tire stock to use, I got a trainer specific tire. The compound is supposed to be better for the trainer(cannot verify that yet since I've only had it for a few weeks) and it definitely is quieter than an outdoor tire.
IF you don't have a spare tire to put in use on the trainer, a trainer tire isn't a bad purchase.