Is thumb-sucking a forbidden habit, and does it invalidate the fast, especially since anything entering the body invalidates the fast?

You must strive against yourself to abandon the habit of putting your finger in your mouth, and remove your finger from your mouth whenever you remember to do so. What happened from you due to forgetfulness does not harm your fast, for the Prophet (peace be upon him) said: "Indeed, Allah has pardoned for my Ummah their mistakes, their forgetfulness, and what they are forced to do."