Yes, and it is routine rather than an exception. Rossville, Fort Oglethorpe, Chickamauga and Ringgold are all regular stops, and cross-border runs are planned into normal Georgia loops instead of being quoted as special trips. This matters more than it sounds: plenty of couriers in this market treat the Tennessee–Georgia line as a zone boundary, which is how a Catoosa County practice ends up driving its own specimens to a Chattanooga lab.
Transport of clinical specimens across state lines by ground does not require a separate license for the courier; what matters is correct packaging, temperature control and documented custody, which apply identically on both sides of the line.