How I've always dried my quilts after washing is to *FIRST* dry them flat until they feel dry to the touch and are about the same weight they were before they were washed.
*THEN* I put the dry quilt into the dryer for about 20 minutes, to fluff it and to make sure the last traces of moisture are gone.
My reasoning is that the seams and fabric in a quilt are most vulnerable when the quilt is wet and has all the weight of the water in it. By waiting until the water is almost completely gone, the quilt is least likely to be damaged by the dryer. I do want to be sure the last of the moisture is gone from the quilt, particularly if I plan to store it after cleaning.
Since I've started reading QB, I've noticed a lot of people who do the reverse--they put the wet quilt in the dryer first, then take it out and let it finish drying flat. Is there some advantage to this method that I'm just not seeing?