Yes an infected tooth can make you sick if the infected tooth or tooth infection gets worse or spreads to other parts of your mouth or teeth.
Tooth infections can lead to swollen gums, and even cause issues with your heart and a possible heart attack.
If you get the tooth infection treated you prevent further infection of any other teeth and prevent other health issues such as heart disease.
If the tooth infection is not too bad then gargling with some Apple Cider Vinegar is usually enough treatment to kill off the bacteria causing the tooth infection.
But if the tooth infection lasts longer than a few days to a week then I would recommend you go see a dentist to treat the tooth infection professionally.
Tooth infections can be serious and cause serious problems if they get too bad.