Sheikh Abdulaziz bin Baz confirmed; May God have mercy on him; There is nothing wrong with men and women washing and brushing their teeth with a toothbrush and toothpaste during the day in Ramadan. But he does not swallow anything, so he washes his mouth and teeth, and spits out what he gets from it, spitting it out, and His Eminence said; Toothpaste does not break the fast, and so does the toothpick, and so does cleaning the mouth with soap or something else. All of this does not break the fast, but a person warns against entering something into his stomach, whether it is toothpaste or something other than paste. In another way, then he spits that out. No, it does not harm his fast. Sheikh Ibn Baz pointed out that whoever deliberately swallowed some of the paste; it breaks the fast, but if something went to his throat unintentionally, or that overwhelmed him, or out of forgetfulness; It does not break his fast.
And the Egyptian Dar Al Iftaa believes that it is permissible for the fasting person to use the siwak throughout the day in Ramadan, and to brush the teeth with toothpaste and brush during fasting, as long as the fasting person is careful not to get some of the paste into the stomach.
Ali Al-Rubai (Abha) @Al_ARobai