Contact dermatitis. If you are asking about a rash around a baby's mouth (not inside) it may be a form of contact dermatitis from newly introduced foods. Sometimes babies will get a red, rough area beneath their bottom lip- this is not an allergy to the food, but just their skim getting used to the new food. Use vasoline before eating to help.
Thrush in the mouth? Milk should not cause a mouth rash in a baby. Thrush is a whitish coating inside the mouth (on the tongue, inside the cheeks, the gums, etc...) caused by an overgrowth of yeast (candida). Candida is normally in people's mouths, but in small amounts so is invisible. Thrush should be evaluated by a primary care doctor, who can prescribe medication if needed (no over-the-counter treatment available).