What happens to eyelashes that fall into the eye and don't get removed?

Not much. They kinda sit there and get encased in mucus which slowly hardens or can cause a granuloma reaction. Sooner or later they become noticeable, and you will dig it out with your finger. Or, you may need an ophthalmologist to dig it out. I have never seen that happen in my experience.
No prob. I have seen eyelashes get embedded into the conjunctiva as a foreign body (long standing). Usually eyelashes falling into the eye (between lid and eyeball) are irritating and thus noticed and removed.
Don't Panic. More often than not, when an eyelash falls into the eye, it creates a tearing reflex which ultimately causes the lash to be flushed from the eye. Rarely, the lash can get stuck between the upper eyelid and the eyeball, which may cause some irritation but rarely anything more. One can attempt flushing the eye with artificial tears but if still irritated, a visit to the eye doctor can resolve it.
Eye feels scratchy. A foreign body like a lash will usually come out with blinking or using some eye drops like artificial tears. They cannot get behind the eye, which is a frequent question.