The red bumps can be due to folliculitis of small hair follicles around the lips and the dry lips can be due to cheilitis. Folliculitis starts when hair follicles are damaged by friction from clothing or blockage of the follicle. Topical antibiotics such as mupirocin or neomycin containing ointment are also useful.In case the symptoms persist then oral antibiotics may be needed. Please consult a dermatologist in that case and get cold sores also ruled out.
It is very difficult to precisely confirm a diagnosis without examination and investigations and the answer is based on the medical information provided. For exact diagnosis, you are requested to consult your dermatologist. I sincerely hope that helps. Take care and please do keep me posted on how you are doing.