Is it obligatory to rinse the mouth to remove food particles that have come up from the stomach and then gone back down or been swallowed unintentionally, or is it sufficient to spit them out and then swallow the saliva?

If a fasting person swallows food that has come up to the mouth after belching unintentionally, their fast is not invalidated. And it is obligatory for anyone who can remove vomit with a basin or by rinsing the mouth to do so. However, the mere persistence of taste in the saliva does not invalidate the fast.