Well recently I've been brushing my teeth a lot like inbetween meals and before bed and in the morning, and the tooth paste would slip off down through my bottom lip and to the chin which is burning right now, but after I brush my teeth I wash my face which I forget to wash my hands after so im touching my face while tooth paste could of possibly got into my hands.
If you can rule out any irritants and cold weather as the primary cause, I would have a doctor quickly check you out. Always best to be overly conservative.
Have you put your hands near your nose & mouth that may have been in contact with something that may have reacted.