Im not so sure about the dirt inside, sure it will be dusty, sometimes muddy, but most of a crews sparetime is spent on keeping the gear i good shape since its vital for your survival.
The driver will look after the engine etc., the radiooperator will take care of radio, the gunner will resighten weapons, the commander will update on intell and the loader will preform a lot of other task, both helping the others, and his own field of work.
All in the crew will help on loading ammo an fuel, but im sure the interiour allso will be looked after - as nobody in the crew has an interest in the loader sliding in a pool of mud in the middle of a battle.
Another thing is the survivability of a tank, will it last long enough to be dirty...
So, the paint will be worn down, in some cases scraped off to prevent flying paintchips like on german tanks, and dust will collect - to a certain point, where the tank will recieve a runover by crew and support personel.