It is generally safe to have an indoor fireplace with a baby, as long as proper precautions are taken. Make sure the fireplace is well-maintained and the baby is not placed too close to the source of the fire. It is also recommended to monitor carbon monoxide levels and consider installing carbon monoxide detectors in your home. Studies have shown that children may experience coughing or wheezing during months when indoor fires are used, but using a fireplace is not strongly linked to respiratory illnesses in infants. However, it is important to be cautious and take basic safety measures to ensure the baby's well-being.