Spock said it well except for the "later" part bwhahaha... Oil will stop flowing on the cams and cam bearings and you will smell smoke "sooner than later".
The way your pump works is very simple. The pump itself is very unlikely to fail since it is so simple and is driven by sturdy teeth right off the crankshaft. What WILL fail is the person who is supposed to put oil in the bike. If the bike uses oil, check the oil level often. The other problem can be debris. The passageways from the bottom where the pump is to the top come up in journals also occupied by engine studs. If you use GOOP (silicon or other stuff of this nature) and it gets into the engine or enough crud gets into the oil, it can clog the oil journals which feed oil to the camshafts and cam bearings. This is why, when you drain the oil, it isn't a bad idea to blow the journals out (when there isn't any oil in the bike) with compressed air and is certainly a required procedure in any rebuild.
ALSO... you DO get sludge and grit in your oil. There is a fine screen over the pump inlet. If you rebuild or just buy a bike, remove the pan and clean the screen. Take a scraper and scrape out the crud on the bottom of the pan and buy a MAGNETIC DRAIN PLUG from z1enterprises.com to catch metal bits that float through the oil because when these get on the screen, they tend to stick there or worse, get into your crank brearings or other places where they can do damage...