This isn't meant to be a complete answer (at least not to my usual standards), but a very good treatment in Straight Dope is worth mentioning. This isn't an act of scientific research, so much as investigative journalism. Cecil Adams (the journalist), did a lot of footwork to get to the bottom of this claim.
An article in the February 18  Wall Street Journal says, "The average mattress will double its weight in ten years as a result of being filled with dust mites and their detritus."
Cecil described his efforts in tracking down the source of the article:
I contacted the Wall Street Journal reporter who wrote the article in question ("Those Costly Weapons Against Dust Mites May Not Be Worth It"). She said she'd gotten this[...] story from a source at Ohio State University[...] I tried reaching Emmett Glass, described in the story as "an OSU research associate leading the university's ongoing Dust Mite Management Study."
Cecil later received correspondence from Emmett Glass of Ohio State University, who is the purported source behind the claim that appeared in Wall Street Journal and exploded into the public eye. Emmett wrote:
I never quoted that statistic. I told [the reporter] that Internet web sites have statistics that try to strike fear in the consumer, thus promoting their products. I gave her a few off the top of my head (two million mites in an average mattress, mattress doubling in weight, etc.) that I read over the years. She asked me if any of these statistics have any scientific merit and I told her that none of them are in the literature. To the layman that is NO! In fact I asked the Wall Street Journal writer to call an expert on mattresses at the internal sleep products association. She did and was told that the statistic on mattresses doubling in weight was far from the truth. The journalist choose to include it in the story anyway. She liked the statistic because it made her story more interesting.
So it would seem that that should set the record straight. It does not explain where the mattress companies originally came up with this flawed statistic - but it's unsupported by scientific evidence, and perpetuated by tabloid journalism and opportunists seeking to profiting from customers’ false beliefs.