Red dots on skin but not itchy? Could it still be allergic? Or some infection?

Rash. Medical advice is difficult to provide for conditions such as rashes because visual inspection is so important to a physician's diagnosis. The distribution of the rash on the body is also important information. The best advise i can give is consult a physician for a history and physical examination.
Allergies itch. Skin eruptions, like the drug induced measles-like rash in the picture, due allergic reactions to foods, medicines, contactants (e.g. Poison ivy) are basically always itchy so if there is no itch i recommend seeing a dermatologist not an allergist.