Is it normal for a person's heart rate to be somewhat elevated when they are ill or have an infection somewhere in the body? if so, why?

Yes: We go into a higher level of metabolism when our bodies are fighting infection. Fever and higher heart rates go hand in hand. It's no big deal unless a patient is so sick that his heart rate stays way up there even though he is getting treatment in the hospital. That scenario would be serious. For most cases though, the high heart rates vary up and down, and settle back to normal as we get well.