Definition. Dust mite allergy is an allergic reaction to tiny bugs that commonly live in house dust. Signs of dust mite allergy include sneezing and runny nose. Many people with dust mite allergy also experience signs of asthma, such as wheezing and difficulty breathing.efinition. Dust mite allergy is an allergic reaction to tiny bugs that commonly live in house dust. Signs of dust mite allergy include sneezing and runny nose. Many people with dust mite allergy also experience signs of asthma, such as wheezing and difficulty breathing.