After our Shih Tzu's face gets wet, whether from drinking water or being outside when it's raining, he will rub his face on our carpet to dry it off... even after having his face groomed. Every now and then he'll sneak into our bedroom and I catch him drying his face off on my pillow. Why did he start using my pillow and how do I get him to stop?

Who knows why he started and why he prefers your pillow?
Get a small towel just for drying off his face. My dog (12 lb JRT) loves her face rubbed vigorously be me at any time, and rubs it on rugs even when not wet.
See if he enjoys a good towel face rub every day! Make sure it's a towel with no hint of soap or fabric softener, and keep it unwashed as long as you can.
And as dogs get older, they get itchier and dryer, and need more brushing and bathing. I bathed my dog every 3 months until she was 11 or 12, but gradually shortened the time and it's now about 3 weeks.
And in cold climates, the dry indoor air just makes every speck of dust and every dust mite more irritating.

He just wants to dry his face, and anything that's handy is good enough for him.

To avoid this, start drying his face for him with a towel when he comes inside, or after he drinks.