Oil certainly is finite; the specific and rare set of circumstances that created our reserves happened millions of years ago. Possibly there might be some in the making now, but I doubt the human species will survive the timeline required for new oil to be made. The theory of peak oil turned out to be a fizzer, at least in the dates predicted. There's quite a lot still there but the problem is the low hanging fruit is running out and what's left undiscovered or is being discovered will be harder to extract and correspondingly more expensive. The ocean regions are very promising except for the logistical side of getting to it. It's not only depth for drilling and extraction, but distance that holds back a lot of deeper ocean development. At present it's not practical for platforms more than 200klm out due to the limitations on servicing by helicopter. With a big price rise in oil, companies would be more inclined to find a work-around to that problem.
With land based reserves there's still promise in some parts of the world, but the day could come when demand outpaces supply. A lot of oil is being found in previously worked out areas or areas that showed low prospect in the past. This is due to the rapid increase in technology of seismic data processing. The pace of change is hard to keep up with at times and previously overlooked areas can be a much better bet only 12 or 18 months later due to better data processing. Most of our exploration data was (and probably still is) processed in Texas and at times it's been like keeping the motor running waiting for their next new trick. I guess that's the way advances in computing are affecting lots of industries around the world.