Does amoxicillin/clarithromycin/omeprazole cause mouth and tongue sores? If so, how do I relieve the soreness?

Oral Sores. You did not mention why you are on these medications. You would have to be seen by dentist and\or physician in order to establish the cause of your mouth and tongue sores and how best to alleviate the problem.
Not usually. Unless you have what is called stevens johnson syndrome in which you have rash and mucous membrane ulceration and it is a potentially fatal condition. If your sores are just canker sores and there is no other constitutional symptoms, try my mother's recipe- put some table salt on it. It will sting for seconds but may well work for you. There is no surefire rx for canker sores however.