When my two daughters with the severe dust allergies were young, the doctor said that their bedding should not be dust catchers, like chenille. Nothing was said about content. I would imagine the quilting lines in a quilt could catch dust. I wouldn't worry about content as much as texture. Either a smooth surface or a quilt you can wash often. I'd go with the smooth on the bed and use the quilt for decor.
As far as batting goes, it is encased in the fabric, so the only time you're exposed to dust is when you handle it. For that reason packaged would be better that buying from a bolt.
The dust covers you buy for the mattress, springs and pillowcases are tightly woven cotton, so unless cotton was an allergen, it's not an issue.
I would wash everything I bring into the house, but I do anyhow.