As far as I know, the purpose of the oxygen sensor in todays systems is purely as an aid to the combustion process?
I mean to say, if the engine is running too rich, then the sensor would let more oxygen into the combustible mixture which enters the injectors, thereby improving the combustion process and also making the engine run better?
Similarly, I would say that if the mixture is not rich enough, then the sensor would let more fuel (all-be-it petrol or diesel) be put through the injectors, in relation to air/oxygen.
So, in essence, I would say all that would likely happen is that you would get a period (lasting until you were able to renew the sensor) of possible rough running, and maybe you using slightly more petrol/diesel than you would normally . . . I would say that not much more than that would be likely to happen.
Other than that, someone else will be along to confirm wether my thoughts are correct.