Wa' laykumusalam wa rahmatuLlahi wa barakatuHu,
It is makrūh to use the siwak while fasting, whether the fast is wajib like Ramadan or sunnah like Mondays and Thursdays. However, this ruling only applies from after zawāl (beginning of ẓuhr) until maghrib. The reason for it being makrūh is the sahih hadith:
The [changed] scent of the fasting person's mouth is more pleasant to Allah than the scent of musk.
The reason why it is makrūh only from after zawāl and not the entire day is due to a hadith narrated by imam Ahmad in which our beloved prophet ﷺ mentioned five blessings that were bestowed upon this ummah during the month of Ramadhan, the second of which is that during the masāʾ the scent of their mouths is more pleasant to Allah than the scent of musk. In Arabic, the time of masāʾ begins after zawāl. In addition, in the beloved prophet's ﷺ saying that 'the scent is more pleasant than musk' is evidence that one should keep this scent, thus removing becomes makrūh.
Allah knows best.
Answered by Shaykh Omar Mohsin