Possibly eczema, which can present this way. Have you ever had eczema or "sensitive skin" elsewhere on your body?
This is unlikely to be an allergy to something touching the skin (lotion, toothpaste), unless you are using something new that touched just your chin and corners of mouth.
Try 1% hydrocortiosne 3-4 times a day for a few days. If the rash goes away or gets much better, that would argue for eczema. If not, then you should see a doctor for peoper diagnosis and treatment. In fact, even if it is eczema, you should see a skin doctor so you can get a prescription-strenght remedy and relevant advice.
Best.
Dr. Rockoff