Praise be to Allah.
It is proven that the Messenger of Allaah (peace and blessings of Allaah be upon him) said: “Snuff water up into the nose deeply, unless you are fasting.” Narrated by al-Tirmidhi (788) and classed as saheeh by al-Albaani in Irwa’al-Ghaleel (935).
This hadeeth indicates that it is not permissible for the fasting person to let water reach his throat via the nose.
Based on that, if the nose drops are small in amount such that they will not reach the throat, then there is nothing wrong with that, but if they do reach the throat and can be tasted, then the fast is invalidated and has to be made up.
Shaykh Ibn Baaz said: The same applies to eye drops and ear drops. They do not break the fast according to the more correct of the two scholarly opinions, but if he finds the taste of the drops in his throat, he should make up the fast in order to be on the safe side, but it is not obligatory, because these are not openings through which food and drink enter the body. But nose drops are not permissible, because the nose is an opening through which food and drink may enter the body. Hence the Prophet (peace and blessings of Allaah be upon him) said: “Snuff water up into the nose deeply, unless you are fasting.” The one who does that has to make up the fast because of this hadeeth and similar reports, if he finds the taste of that in his throat. End quote.
Shaykh Ibn ‘Uthaymeen said in Fataawa Ramadaan (p. 511): If nose drops reach the stomach or the throat, then they break the fast, because the Prophet (peace and blessings of Allaah be upon him) said, according to the hadeeth of Laqeet ibn Sabrah: “Snuff water up into the nose deeply, unless you are fasting.” So it is not permissible for the fasting person to put drops in his nose that can reach his stomach or his throat, but as for nose drops that will not reach them, they do not break the fast.
With regard to eye drops – and also using kohl – and ear drops, they do not break the fast of the fasting person. End quote.
Based on this, the fasting person should not use nose drops unless it is too difficult for him to go without them, in which case he should use them but he should be careful not to let any of that reach his throat. If he swallows any of that he should make up that day.
If he knows that he cannot help swallowing some of them, then it is not permissible for him to use them unless his sickness is so bad that it is permissible for him not to fast, which is if fasting will cause him harm or unbearable hardship. See also questions no. 50555 and 38532.
And Allaah knows best.