If they are skinny road tires, stay of the grass etc. The smoother they are, the more important this rule is. The smoother they are, the more likely it is they will pick up debris - particularly if the ride surface is damp or wet.

I follow this rule to the extent on my road race bike, or my time trial bike, (which both have very smooth tires) get carried /lifted to the road, and not wheeled on grass verges or curbsides.