late for (school, class, dinner,...)

late to (school, class, dinner,...)

Are be late to and be late for interchangeable?
You can use either one.
Use to if you're thinking of school, class, or dinner purely as destinations.
Use for if you're thinking of being somewhere for the purpose of participating in the activities characteristic of the destination.
The second of the two choices is probably what most people are thinking most of the time when they are late, so I suspect you'll hear for more often than to.
Pretty much, except it depends on the verb you use. For example: I'm going to be late for school and I'm going to be late to school are pretty much both correct, though as a native speak, the former sounds better to me than the latter. Another example: I arrived late to school is more acceptable. I arrived late for school doesn't work as well.
 CalifJim's reply was promoted to an answer.