Actually, this was more of an academic question, on a friend's behalf. His "Check Engine" light illuminated at 55K miles; dealer service at 61K miles indicated that the oxygen sensor was bad. Other than the warning light, the vehicle is asymptomatic (no change in fuel economy or performance) and his state has no emissions inspection, so he's felt no pressure to renew the sensor.
As for theoretical consequences, decreased fuel economy and increased pollutants come to mind; additionally, extended running in a full rich condition would eventually damage the catalytic converters. Any other harmful effects related to a failed sensor?