Should I regularly open the flow to all the pool lines?

My pool has a spa return, pool return, fountain, skimmer, vacuum, and main drain. Sometimes I don't use certain lines very often, like, say, the fountain line. It might be a month or more before I need/want to use that line, so the water in that line doesn't get circulated regularly.

Is there a correct protocol for how frequently one should circulate the water in the pool's inflow and outflow lines, so something nasty doesn't start growing inside the pool line?

I would say run each line as least as often as you test the water and/or maintain the pool. Once a month may be pushing it but once a week should be enough. But one trick is to open the line just a tiny bit so water is always flowing through the line albeit at a very slow rate. Then you don't have to worry about it.