What are the common administrative tasks related to networking that require root for networking? All I can think about is stuff like route tables and dhcp, both of which live at the IP/Ethernet level rather than TCP.
Starting any network server process that uses ports under 1000, like most standard protocols (https, http, ssh, smtp, dns, ntp, dhcp etc.), requires root rights on any UNIX-like operating system.
Most personal computers do not need server processes (unless you want to connect remotely to them), but your question was not restricted to them.
From a practical point of view, regardless of the scope of the original question, this is the kind of scenario where you'd really want the restriction. More than a simple administrative task it's a dangerous attack vector to allow any user to launch your httpd or DNS.
That being said, check out capabilities(7) in Linux. You can grant an executable the privilege of binding to a low port when run by non-0 uid through setcap. This is a good compromise.
this whole 'privledged ports' nonsense is left over from a time where some process on another machine running on a low port was somehow to be trusted - because the person running that process was another administrator, and you can generally trust those guys (as opposed to unwashed users).
that world didn't last very long, and I wish we could vent some of these designs that didn't pass the test of time.
You can adjust the highest priviledged port (at least on FreeBSD). It's convenient to set that to 79 and let regular users listen to http without needing root to listen.
ssh and smtp generally need root to do their job, although maybe you could find a way to deliver mail to users without it. If you want to run user based dns or others, you could set the priviledged port even lower.