From islamtoday.net. The link can be found here: Using toothpaste while fasting
However, without going so far to say that it is makrûh to use toothpaste while fasting, Sheik Ibn al-`Uthaymîn discourages it. He says:
There is nothing wrong with a fasting person using toothpaste, as long as it does not go down his gullet. However, it would be better to refrain from using it, simply because it has a strong tendency to go down a person’s throat while he is unaware.
For this reason, the Prophet (peace be upon him) said: “Exaggerate in inhaling water (while performing ablutions) except when you are fasting.”
Therefore, it is better for a fasting person to refrain from using toothpaste. The matter is easy. If he simply waits until after he breaks his fast to brush his teeth with toothpaste, he will have avoided what might possibly nullify his fast.
And Allah knows best.