These acts are allowed in general, but it is disliked to exaggerate (that is, use a lot of water and put the water deep into the mouth or nose while fasting). Laqit ibn Sabra reported that the Prophet said: “Exaggerate when rinsing your nose unless you are fasting.” This is related by an-Nasa’i, Abu Dawud, at-Tirmizhi, and Ibn Majah. At-Tirmizhi called it hassan sahih.
Scholars dislike using nose drops (that is, applying medicine through the nose) while one is fasting, for they are of the opinion that it breaks the fast. There is a hadith that supports their opinion.
Ibn Qudamah sums up the various opinions on the subject: “If while gargling or rinsing the nose for the sake of purifying one’s self [for example, for prayer] water reaches the throat unintentionally and not due to exaggeration, there is no problem. This is according to al-Auza’i, Ishaq, and one statement from ash-Shaf’i, which is related from Ibn ‘Abbas. Malik and Abu Hanifah hold that it breaks the fast because that water reaches the stomach. If he was aware that he was fasting, it breaks his fast, as if he would have drunk intentionally. The first opinion is stronger, since [the water] reached the throat without intention or exaggeration. It is similar to having a fly enter the mouth and proceed to the throat. That differentiates it from an intentional act.”