It will negatively affect your gas mileage by a lot.
It is the O2 sensor that allows the computer to run on the edge of engine destruction (engine knock). Without it the computer has to play it safe and run rich all the time. This is why the older cars used so much gas (or the newer cars have improved so much on MPG).
Also it depends on which O2 sensor failed. The downstream O2 sensor just checks if the Cat Converter is working or not.
Good Luck...