Yes Dental Abscesses can make you feel sick and if you're feeling sick from a Dental Abscess you need to see a doctor or your dentist as soon as possible about it before it gets worse.
Dental Abscesses that are causing you to feel sick such as fever, chills, nausea, vomiting, or diarrhea should be checked out by a doctor and then see your dentist.
Tooth problems can lead to other more serious problems so always get your teeth looked at especially if you're feeling sick from A Dental Abscess.