Of course, it would depend greatly on what your source of running water was: a local groundwater aquifer, a nearby lake or river, a long pipeline from a distant source, a de-salination plant, etc.
No matter what the ultimate source most places use water towers and/or reservoirs to maintain water pressures. They use pumps to fill tanks on high points and then let gravity feed maintain the pressure. As long as the pumps keep working you will have some water, but most places don’t have fully automated pumping systems. As such we can assume water will stop being pumped within a matter of hours.
Once the pumping stops your water supply is restricted to how long the water supply in the reservoirs lasts, and that in turn depends on how fast it’s being used. If people turned into zombies gradually then they may well have time to shut down many water outlets and just as importantly shut down many fire risks. As workers get sick factories and so forth will be shut down and the water largely turned off and the same for most household water use. If people instantly come over all zombie the water will simply continue to flow at an approximately normal rate as people wander away from workstations and showers to seek brains. Also in that case many fires, irons, stoves etc will be left unattended and cause fires which in turn will produce ruptured plumbing and uncontrolled water flow.
Assuming gradual zombification and minimal fires water could last for several days to a week before open faucets of various kinds leaked away all the water in the reservoirs. In the case of rapid zombification you’d probably be sucking the faucet for a drink within a 6 hours…