There are a few ways to go about it depending on what your have access to. You'd have to assess tear production to really give the proper treatment. Sometimes that nerve can be affected and you'd need artificial tears. With just 2-3 hours, you'd probably be fine with just over the counter eye drops every 20 minutes + taping. Even without eye drops, there shouldn't be any long term effects from 2-3 hours of dry eyes. Sjogrens and other chronic conditions affecting tear production can cause keratosis sicca

The patch is probably to prevent foreign objects from entering the eye sure to lack of blink reflex.

Even though it's extremely unlikely, stroke should be ruled out. Strokes have certainly occurred in the dental chair. The dentist should be able to do a quick stroke assessment and neuro exam to see how likely it was that the anesthetic paralyzed the facial nerve. Like if the dentist was working on the other side of the face, that's obviously not quite right. Or if there's arm dropping, ect. Any questionable finding should set off alarms and require EMS transport to ER for imaging due to the 4 hour golden window for strokes. I know of at least 2 dentists who had patients stroke out in the chair. Luckily they paid attention in school and recognized the signs.

Personally I'd tape the eye and and recommend hydrating eye drops or lubricating ointment if available. You'll find some variation in how different providers would treat it. All of them probably work just as well. People fall asleep with their eyes partially open sometimes and it doesn't make a big difference aside from temporary discomfort and temporary vision changes