My boyfriend got a burn on his arm an developed a rash how do we treat it?

Probably see an MD. A "rash" after a burn is suspicious for an infection, which needs to be treated with antibiotics. Otherwise it could be a dermatitis, which is treated with topical medicine. It's hard to know in a forum like this, but i would certainly get it looked at.
Not sure. It's difficult to say. If he was burned and then developed cellulitis, he would require antibiotics. If he healed from the burn and now has a rash of the newly healed skin, gentle cleansing and non-comedogenic moisturizers might be appropriate. It should be looked at by a surgeon who treats burns before beginning treatment.
Google it. A rash after sun exposure is not that uncommon, especially if it is the first in a while. Usually self limited, if it persists you should see a doctor; steroids sometimes helpful. May try otc Hydrocortisone for a few days.
Depends. Treatment depends on a diagnosis and there are many causes of rash, but this is likely due to over moisturizing of the normal skin around the burn or a fungal rash from use of the antibacterial ointment. Either way try to keep the normal skin more dry and it may resolve.
Allergic reaction. It's possible the rash could be an allergic reaction to topical antibiotic pin entrust was placed on the burn, if the rash is localized to the area around the burn. I have seen this many times with neomycin ointment.
Thermal or rug burn? It depends on the type of burn and the depth. Topical antibiotic over the counter antibiotic, if not allergic, may be all that is required. If deeper medical attention should be obtained, .