No, this is not a good idea.
Lots of things are within the scope of this site, but aren't operating system-dependent. For example: all the hardware questions, router setup (many concepts aren't firmware dependent; port forwarding is essentially the same on every router, even though the mechanics of configuring may be a bit different).
Tagging these questions with the operating system would make them less useful, because it would encourage people to say the exact steps to follow, rather than say what you'd actually be doing. Continuing with port forwarding: which is more useful? "Set up port forwarding", or "Click this button, then that button, type
foo, save settings"? The former option doesn't make the user look up the exact make of their router, and gives them a guide for how to solve these problems in the future. If they want to know exactly how to set up port forwarding, they can ask a new question, but there's no reason to bring that into play when they want to just set up a game server.
Also, remember something: The OP often doesn't know the cause of their issue; that's why they have an issue. They might not know if something is OS-dependent or not. If they tag incorrectly, it either restricts the possible audience (someone who knows how to solve the problem might think it depends on an OS they don't know), or winds up in front of people who have no idea how to handle it (even though the question says it is OS-independent, it isn't).
Lastly, the OS-independent tag wouldn't be useful on its own; it'd be more of a meta tag (what would a question look like if the only tag that applies is "OS independent"?). This is discouraged on Stack Exchange sites; tags should work even if there are no other tags on the question. In this case, it'd also take away one of the 5 tags for something completely useless - saying that operating system doesn't matter.