Is my baby's thrush causing my nipples to be sore?

Yes. The germ of thrush can pass back & forth between baby & breast and cause inflamation in the milk ducts and a dermatis on your skin. Often drs use the same Nystatin suspension as a nipple dressing if this is happening to help.Other treatments are available. Check with your provider & discuss the issues.
Not Likely. . Thrush thrives in a warm moist environment in an immunocpromised patient or a patient on antibiotics. It is generally asymptomatic and not painful. It is generally not contagious. Look for other causes.