As for using abrasives on a lathe, this is less of a concern on a wood-turning lathe, but can be a serious problem on a metal-turning machinist's lathe. It's not the sawdust, which is usually only a nuisance and easily cleaned up unless it gets into oiled bearings, in which case it turns into gunk that has to be washed out with solvent. The real problem with using abrasives on a machinist's lathe, particularly, is not the sawdust, but rather the bits of abrasive grit which may even be microscopic. This abrasive grit will collect on the usually oily surfaces of a metal-working lathe and cause severe wear on the various working surfaces which will eventually destroy the high-tolerance accuracy of the machinist's lathe in a fashion likely impossible to repair short of regrinding all the bearing surfaces to restore the previous accuracy of the lathe, if that is even possible. And, no, a machinist's lathe is not "just a lathe," unless you don't mind starting in the four figures paying to replace one.
One useful prophylactic I've found is to cover any surfaces which may pick up abrasive material with kitchen tinfoil to serve as a cover for the lathe parts. I would not advise using cloth or other flexible material because this poses the risk of being picked up by the moving parts of the lathe with potentially disastrous results. Forming the tinfoil around the covered parts, and securing it with tape if needed, should shield the lathe from the abrasive grit, which can be cleaned off the tinfoil or removed with the tinfoil if one is careful doing so.
A decent machinist's lathe that is respectfully cared for and maintained should last several lifetimes. A relatively new machinist's lathe that has been abused is often not worth much more than scrap value to anyone wishing to do accurate work with it.
