Can viral pneumonia give you a rash?

Yes. Many types of viruses can cause pneumonia and not infrequently they may also cause a rash. The classic is varicella virus which is also known as chicken pox. Adults with varicella seem to be more predisposed to getting viral pneumonia in association with the classic rash.
Not typically. The kinds of viruses that cause pneumonia in adults are not typically associated with a rash. There are some viruses that can cause both pneumonia and a rash, these would be more common in children or adults whose immune systems are not normal. A classic example would be a pneumonia associated with the chicken pox virus.