Does Mouthwash & toothpaste need to be Kosher for Pesach?
Mouthwash and toothpaste are not only a concern regarding Pesach but also the entire year as some may contain animal-derived (not kosher) ingredients.
See here: Halacha2go.com article #589: Does toothpaste have to be Kosher? מ”מ
Therefore, especially regarding Pesach when a person should be extra careful about what he puts in his mouth (or in some cases where he ends up consuming the item, even though it’s probably unintentional), only mouthwash and toothpaste with a reliable hechsher (or on a kosher list) should be used.