Since public universities are nonprofit organizations, do their employees including faculty pay less taxes than those who work in industry?
While I don't think this is true for any universities, the employees of certain non-profit international scientific organisations, including CERN, EMBL and ESA are exempted from national taxation. They are instead required to pay an internal tax which returns a percentage of their income to the employing organisation, with a tax band structure similar to conventional income taxes. In some cases this might result in an employee paying less than if they were under their national tax regime, but it might also result in them paying more.
In America, employees of governmental or nonprofit organizations are taxed the same way as employees of for-profits. The non-profit itself is generally exempt from taxation, but its employees are not. So there are no obvious tax advantages to working at a public or nonprofit university versus working for a for-profit corporation.
(I assume this will be much the same throughout the developed world.)
Here is one case in which this is sort of true.
I work for a public university in the US state of Colorado. Colorado has decided that its public employees shall not participate in the federal Social Security system (states have the right to do this, under US law, though not all actually do so). As such, employees of public universities in Colorado do not pay Social Security tax. However, this means we also do not earn credits toward receiving Social Security retirement benefits. (The state used to offer its own pension system to state employees, but this is now being phased out in favor of a third party-managed defined contribution scheme.)
(We do still pay state and federal income tax at the normal rates.)
So this is one case in which university employees are exempt from a certain tax. But it is not related to non-profit status (private nonprofit universities in the state do not have this exemption) and it comes with a corresponding loss of government benefits.
Perhaps more directly related: I believe that at all US universities (all states, both public and private), student employees are exempt from paying Social Security tax (and from earning credits). But this does not universally apply to full-time employees.
On an international note, in Germany, university professors are considered to be in a special class of public employees known as Beamter, and are exempted from all federal payroll taxes except ordinary income taxes and the reunification tax. This also means, as indicated by Nate Eldredge in his answer, that they are not able to participate in the federal social insurance scheme. Instead, they receive a pension paid for by the individual state governments in which they work.
Another "partial" case. In some states, when a teacher buys supplies for the classroom paying from her own pocket, she can be exempted from paying the sales tax on those items.
50 years ago I had a summer job at a federal lab. One of the old-timers there once joked about a time (even then it was "long ago") when federal employees were exempt from federal income tax.