Is interstitial lung disease an allergic or immune problem?

Both and others: Interstital lung disease refers to the supporting structures, tissues and cells supporting the alveoli of the lung. When this is inflammed or scarred the thickening reduces oxygen-co2 exchange. There are many possible causes for ild and the two you listed are possible causes.

Usually immune: There are some interstitial lung diseases caused by allergies to mold and bird droppings, but most are caused by an abnormal immune response in the lung. There is an abnormal repair process that leads to fibrosis ( scar tissue) which replaces normal lung tissue. Hypersensitivity pneumonitis is an interstitial disease caused by an exaggerated immune reaction to foreign protein that is inhaled.

Depends on the type : Interstitial lung disease is a generalized term used to describe multiple diseases that affect the interstitial tissue (the lung tissue among the airways). Some of them can be related to immune problem esp., if the patient has autoimmune disease such as rheumatoid arthritis, or lupus etc. Some of them related to allergy and the others are idiopathic ( we do not know what causing the disease).