late for (school, class, dinner,...)
late to (school, class, dinner,...)
Are be late to and be late for interchangeable?
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.
CJ