Is it permissible for the fasting person to use toothpaste while he is fasting during the daytime of Ramadhan?
There is no harm in that as long as one is careful not to swallow any of it. Likewise, it is legislated for the fasting person to use the tooth stick at the beginning of the day and its end. Some of the people of knowledge held the view that using the tooth stick is disliked after midday, but this is an incorrect view. The correct view is that it is not disliked due to the generality of the Prophet's statement: The tooth stick is cleansing for the mouth and pleasing to the Lord.  This was recorded by An-Nasa'i with an authentic chain of narration from A'ishah -May Allah Be pleased with her-. It is also due to his statement-Sallallahu Alaihi Wasallam-: Were I not placing a hardship upon my Ummah, I would have ordered them to use the tooth stick for every prayer.  This Hadith is agreed upon (By Al-Bukhari and Muslim) and this includes the Zuhr and Asr prayers, which are both after midday. And Allah is the Giver of success.
1.An-Nasa'i no.5. 2.Al-Bukhari no.887 and Muslim no.252.
Shaykh `Abdul-`Azeez Bin Baz
Fatawa Islamiyah Page no. 262 Vol: 2