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Without examination a confirmed diagnosis cannot be given. Your symptoms could be due to stomatitis or due to angular stomatitis. Irritation and fissuring in the corners of the lips is termed as angular stomatitis. It can also extend to lips and area around it.
Most common causes are iron deficiency anemia, or vitamin B deficiencies. Try taking some iron supplements and multivitamins for some days and see if the symptoms persist.
They can also be caused by allergies and infections. You could either be suffering from contact dermatitis or from allergy to some food. You should get an allergy test done, either a skin test or RAST, to know what you are allergic to. Contact dermatitis could be due to tooth pastes, mouth cleansers, spoons, cups, glasses, food items, cosmetics, shampoos, etc, basically anything that comes in contact with your lips.
Since there is burning in the nose it can also be associated allergic rhinitis.
It can also be an extreme case of gastric reflux causing soreness in mouth and burning of nasal mucosa.
Please discuss this with your doctor as a confirmed diagnosis cannot be given on net. Take care!