If you make the pots out of newspaper (black & white only is the best -- definitely not glossy/coated), and the plants aren't overgrown, then just plant the pots in the ground. Newsprint will rot away quickly.
If the plants are large (starting to become root bound in the pots), it wouldn't hurt to slash the paper a bit when planting so that the roots don't have to do any work to push through the paper.
I haven't used paper pots, but I'm guessing that with the watering you do during seed germination and the first few weeks of growing the paper will be well on the way to disintegrating away by the time you get around to transplanting.
The advantage of paper pots versus plastic pots is that you don't have to remove the seedlings to transplant. This avoids root disturbance which sets back seedlings after transplanting. I suspect the advantage of paper over peat is that paper pots will rot much faster than peat pots (I've seen peat pots persist quite a while after transplanting, especially if the pot isn't slashed up when the transplanting is done).