My first question - Why on earth are you using 'drying towels' for applying tire shine? Drying towels are for drying.
.. unless, as RTexasF pointed out, they get old and grungy, the relegate them to dirty tasks, like tire shine, wheel cleanup, etc..
IMHO - For anyone who wants to seriously approach maintaining their car's appearance, you buy towels for specific tasks, and only use them for those tasks.
- High-quality MF towels for polish removal
- High-quality drying towels to dry the car
- Glass-specific MF towels for cleaning glass
- Decent-quality MF towels for maintaining the interior
When it comes to other stuff, that is more.. grungy.. I tend to buy bulk packs of terry cloth towels, or big packs of the cheaper MF towels (the ones from Sam's Club are great for this).
Also, if any of my nicer towels get to the point where I feel they are no longer suitable for 'finer' use, such as if I feel a high-quality polish removal towel might be at risk of scratching my paint, I relegate it to the lower tasks..
.. Higher-end polish-removal towels get moved to interior or chrome wheel cleanup
.. Interior towels get moved to wheels, doorjambs, etc..
THIS VIDEO HERE is one of the best videos about MF towels, and offers a very good approach to 'rating' your detailing towels, when to stop using them, etc..
... Skip forward to 18:00 in the video to see where he talks about the "rating" system.
Yes, he's really OCD about this sort of thing, but it's his job, and he really knows what he's doing.. and honestly, towels are fairly cheap and paint can get expensive, so I think he's right on point.
--- Bottom line on your towel - If it's a drying towel, but you got tire shine in it, then it's now a 'Tire shine towel.' If it were mine, I would never even think about using it to dry my car any more.