Is dermatitis very contagious?

Depends on what kind. Depends on what kind of dermatitis. That's a very nonspecific term for any rash of the skin. Sone rashes are infectious in nature such as tinea corporis (ringworm) or impetigo (staph infection).
Depends on cause. Eczema or dermatitis are general terms we use to describe a group of inflammed skin conditions that causes chronic, itchy rashes. Types of eczema include: allergic, contact, irritant, nummular eczema, and many others. For the most part, these are not contagious but rather may involve a person's overactive immune system in the presence of certain conditions, materials or allergens.