Can an illegal immigrant who overstayed their tourist visa by a year apply for the green card lottery while they are living in the US?

No. He or she is not eligible for permanent residency. The foreign national is now unlawfully present in the US and must voluntarily return to his or her country of origin or be officially deported. Because the foreign national remained in the US unlawfully for more than one year, he or she cannot apply for reentry into the US for 10 years.