Your description sounds typical of angular cheilitis, which is a fairly common inflammatory condition affecting the corners of your mouth.
It can be caused by several things, eczema being one of them. It could also simply be due to dry, chapped lips.
Occasionally, it can be caused by a type of infection due to bacteria, yeast, or virus.
Depending on the underlying cause, it may persist indefinitely if you don’t seek help from a doctor. GPs will be able to start you of with the right treatment, and they may refer you to see a dermatologist if necessary.
As a start however, frequent moisturising of your lips with Vaseline, and NOT rubbing will help the redness.