I think that at most rinks, private lesson figure skating instructors are private contractors who set their own rates, and what they expect to get is included in their rates, so tips aren't expected.
Tips are most common in lines of business where the person giving service gets paid very little (maids in hotels, waiters at restaurants, cab drivers working at regulated rates, etc.), which isn't true of paid figure skating instructors any place I know of.
Examples of instructors who sometimes aren't paid much are surf and ski group (and sometimes private) lesson instructors who work through schools that set their rates for them. A tip might be appropriate, especially if they do an unusually good job.
If a private coach offers a group lesson, without an intermediary school paying the coach's salary, and many fewer students than the coach hopes for sign up, the few students who do might decide to give the coach tips, to help make up the difference, if they don't want the coach to discontinue the lessons.