Is it ever ok to wash a towel that has had mineral spirits on it?
There are 2 separate prongs to this problem.
Allowing flammable solvents into machines and sewer systems.
We don't want to wash towels that are contaminated with flammable solvents, because the vapors will wind up all sorts of places we don't want flammable solvents.  That includes:

Flushed down into the sewer pipe - sewer gas is bad enough!
In the mechanisms of the washing machine where it could attack plastic components and potentially start a fire.
Wafting through the dryer where it will make contact with hot heating elements or worse, gas burners!

Certain flammables will self-combust
That is because they have a &quot;curing&quot; behavior similar to epoxy or concrete - and like most cures, it is exothermic.  If that heat is sufficiently bottled up, it can rise to its own flash point and FWOOF!  About 1700 house fires a year are caused by this.  That generally relates to paints and stains.
However, I generally recommend developing the habit of handling all &quot;oily&quot; rags on the assumption that they will behave exothermically.  I spread them all over the shop for a week to dry out, or better, leave them out in the rain.