I inherited my father's 2003 xj8 and the front upper shock bushings are deteriorating. Would I be causing damage to the suspension of I drive it in this condition?

 

Is your car the previous all-steel XJ8, (X308) or the new aluminium XJ8 (X350) ? The earlier car has separate shocks with upper bushes, the later car hasn't. So if your car is the X308, (as I suspect), then it would be best to replace these bushes as soon as you can, but the rest of the suspension won't be damaged. Only thing that might be is the body locating area if you let the bushes get so bad you get metal-metal contact.

Best to put more questions in the X308 forum if you have the all-steel car.