بسم الله الرحمن الرحيم
It is makrooh to use toothpaste whilst fasting. One should avoid it. Though it does not nullify the fast but it decreases the reward.
وكره له ذوق شئى وكذا مضغه الخ (درمختار)
If the particles of the toothpaste will enter into throat then fast shall be invalid. (L)
Allah knows Best!
Darul Uloom Deoband