I accidentally got some toothpaste in my eyes. Would anything bad happen if I didn't wash it all out?

Rinse ... Rinse with eye wash to get the abrasive out. And there may be a short irritation from the acidic ph if present.
Eye irritation. Unless you have active gum or teeth infections you should be ok. If you have either of these conditions you should seek out topical antibiotic treatment to prevent potntial complications.
Wash eye out. Most toothpaste have bleaching agents in them. This is usually a hydrogen peroxide chemical and will not damage the eye. You should still wash out the eye, this will help your eye fell better sooner than later.
Yes. It would be important to wash or rinse out your affected eye(s) as thoroghly and soon after the incident as possible. Some ingredients in the paste could easily irritate the eyes.