Neither the state nor the county clears snow and ice from public walkways (sidewalks and trails). While not legally obligated, residents and businesses are asked to help keep sidewalks safe. They should, as soon as feasible, clear snow off the sidewalks in front of their property so that all pedestrians, especially school children, those with disabilities and the elderly, may walk securely. Homeowner associations may require members of their communities to clear walkways abutting their property. Please contact your association or property manager for further information. Read more about snow removal responsibilities in Fairfax County.
http://www.fairfaxcounty.gov/dpwes/navbar/faqs/snowremoval.htm
No official laws in place, but an HOA may impose rules. Would need to be in their bylaws though most likely. The head of the HOA can't just demand people remove snow out of personal preference.