Lying is a major sin itself however if someone takes a False oath on Quran then sin is greater, person has to do isteghfar and make sure not to repeat in the future.

You don’t need to pay kaffara
Kaffara is wajib in following situation

Ruling 2687. If a person takes an oath to do something or to refrain from doing something – for example, he takes an oath to keep a fast or to stop smoking – then in the event that he intentionally (ʿamdan) does not fulfil his oath, he will have committed a sin and he must give recompense (kaffārah). That is, he must either free a slave, or feed ten poor people (fuqarāʾ), or clothe them. And if he cannot do any of these, he must fast for three consecutive days.