What happens if rinse with white tartar control pre-whitening rinse for too long?

Depends on...: What do you consider to be too long? Typical rinsing for about 1minute, if you were to rinse for a very long time, possibly you could develop short term tooth sensitivity. Do not swallow the rinse.

Could irritate gums: Without knowing ingredients, hard to say. Many rinses contain alcohol, which is drying to soft tissues, and overuse should be avoided. Bottom line - rinses will never effectively whiten teeth, and tartar can most effectively be avoided by brushing and flossing well. Rinses are not the answer. So best not to overuse rinses.