Why do older people's nose run when eating?

I've not seen that this is the case. Maybe the person you're speaking of has a cold.

Histamine is a chemical in the body involved in immune response and gastric secretions as well as other functions. It causes people's noses to run when it dilates blood vessels along the mucosa causing the capillaries to leak. It's likely that the increase in histamine from eating due to involvement with gastric secretions also triggers the nose to run.
Why exactly it happens more in elderly people is a question for a geriatrician, though as with many other bodily functions, age alters the normal mechanisms.
If you want more detail, look up histamine as a chemical and the role of histamine in gastric (stomach) physiology and its triggering of runny noses.