This has been discussed here before, but, since the claim that drivers pay for the roads and cyclists don't is so pervasive, it's worth repeating: cyclists more than pay for their fair share of the road. This is backed up in great detail by US-PIRG (United States Public Interest Research Group) in the latest version of their "Who Pays For Roads" report. Spoiler alert: not drivers.
A 2014 analysis by Advocacy Advance of statewide transportation improvement programs (STIPs)—short-term, fiscally constrained plans of transportation projects required of states under federal law—found that less than 1.5 percent of all funds were programmed for bicycle, pedestrian or shared-use projects
Bicyclists and Pedestrians Already Pay for Most of the Roads they Use
As general taxpayers in their communities, people who walk and bike help pay for the maintenance of streets, which are predominantly dedicated to the storage and movement of motor vehicles.
The degree to which urban streets are dedicated to automobiles is illustrated by a 2014 analysis of the use of roadway space in San Francisco, one of the least auto-oriented cities in the United States. The study found that 71 percent of all paved road area within the city was devoted to general traffic lanes geared primarily toward the movement of cars. An additional 11 percent was devoted to freeways (which are automobile-only) and state highways, and 15 percent to on-street vehicle parking. Only 2.4 percent of street space was devoted to transit-only or bike-only lanes—this, in a city in which private automobiles account for fewer than half of all trips. Thus, a San Francisco resident who does not use a car would pay most of the levies that support the city roads while using only a tiny portion of that infrastructure.
Bicyclists and Pedestrians Impose Negligible Roadway Costs
Compared with automobiles and trucks, pedestrians and bicyclists impose little wear and tear on road surfaces.A general rule of thumb is that the damage a vehicle imposes on a road surface increases to the fourth power of axle weight—that is, a vehicle that weighs ten times as much per axle imposes ten thousand times as much roadway damage as a lighter vehicle. A 200-pound bicyclist with a 50-pound bike, therefore, will impose approximately 1/65,000th the roadway damage of a 4,000 pound car.
Bicyclists and pedestrians also take up little room on roads. A stationary pedestrian takes up one-80th of the space of a parked vehicle, and a bicycle one-20th of the space. Compared with a vehicle traveling 60 miles per hour, a pedestrian takes up one-250th of the space, a bicyclist one-100th of the space, and a bus passenger one-67th of the space.
Estimates of the external costs imposed by walking and biking validate the conclusion that it is inappropriate to charge bicyclists and pedestrians user fees. A 2009 analysis by the Victoria Transport Policy Institute estimated that the external cost of a mile of bicycling was less than a penny, while the cost imposed by a mile of walking was 0.2 cents—compared with external costs of driving of more than 29 cents per mile.
Even if people who bike and walk were to be charged fees based on the impacts of their behavior—something that has never been fully required of drivers—those fees would likely be so small as to be barely worth collecting.
All of this has been mentioned here or in the comments before, but it's nice to have some numbers here. More - and citations - can be found in the report. Bookmark it for your Washington Post-based comment arguments.
Here is the 2011 version, for those who are interested.