I just saw on Fifth Gear (a UK TV car show) that it was illegal to drive a dirty car in Belarus. A Google search revealed many other people believe this to be true.
Can anyone confirm it is true and explain why?
It is true, but in a very unspectacular way.
10. The driver is prohibited to:
10.3 participate in road traffic on a vehicle covered with dirt (laminations), limiting the driver's field of vision and also making indiscernible the information on the registration plate at a distance of 40 meters and less
So - yes, you can be fined (up to 300,000 Belarus Rubles, that is, up to about 30$) for having a dirty vehicle - but only if you can't see the road or the police can't see your licence plate. This is legislation the kinds of which have been seen in many a country.