No they don't
The CDC has a helpful page explaining how to properly repel mosquitoes to prevent the spreading the spreading of the West Nile virus.
One would expect to see dryer sheet mentioned there if they worked. They aren't. Unsurprisingly, they only mention commercial mosquito repellants and pesticides.
CDC evaluation of information contained in peer-reviewed scientific literature and data available from EPA has identified several EPA registered products that provide repellent activity sufficient to help people avoid the bites of disease carrying mosquitoes. Products containing these active ingredients typically provide reasonably long-lasting protection
Also the EPA states that:
Before you use any insect repellent, be sure you read the label for any warnings and also to see what the active ingredients are. The substances in insect repellents that actually repel the insects are the active ingredients, and are registered with EPA.
In other words: mosquito repellants must have a specific "active ingredient". What is the active ingredient in this case?
Snopes also dismisses the claim as absurd.