Gun control debates: Nearly always take place between people who have guns, enjoy having guns, and don't see a problem with people having guns as a result and people who are afraid of guns and don't think people should own guns. No progress between these groups can be made as the discussion rarely moves beyond the two sides' fears and into facts or considerations of various forms of legislation or constitutional matters. One side is afraid guns will be taken away, the other is afraid people have guns. I rarely see facts that are clear and decisive pushed forward, and very, very rarely see plans of action that could be considered under current legal interpretations and put into action, or would even work (eg, "assault weapons" bans, which are useless).
Theological debates: nearly always take place between religious people who are unschooled in theology (or philosophy/logic) but attached to their beliefs and secularists who enjoy pointing this flaw in lacking knowledge and well-thought-out reasons out. This is not because secularists can not or will not debate people who actually know something of theology. It's that there aren't very many people who know something of theology (or logic). Most people appear to believe things that they're told to rather than digging into it very deeply or credit subjective experiences and interpretations/perceptions very highly as evidence rather than dogma. Arguing about dogma therefore is a waste of effort for most such discussions as a result. The logical pretzels are very stale and can be seen coming from miles away. I get very tired of seeing Pascal's Wager come up. Pascal is an idiot on this point.
Libertarians vs other libertarians: Lots of holier than thou digressions occur. Various sources of internal worship are established and battle lines are firm. Most of the discussions are pointless anyway since most people are not very libertarian in their political thinking or political philosophy and it results in a lot of theory over reality. Very little of pragmatic advice will emerge that can be sensibly applied to the real world as it is right now (as opposed to the dream world where libertarians somehow rule over a world with various problems willed away as having been fixed somewhere). Libertarians should basically only talk to "normal people" and learn how other people think as a result (probably wise counsel for any political grouping is to talk to people that aren't in much agreement and find out why). I admire some of the theoretical work of libertarian political philosophy. I don't admire the "steal underpants - ??? - profit" mentality at times as an approach to political reform.
What I would define as a doomed conversation:
1) A conversation that will not change either participant's mind. Nothing can be learned (by either party) and no opinions will be amended.
2) A conversation that follows a certain predictable script of "here's point A that always comes up, here's point B".
3) If it follows a certain script, nothing will be learned, no minds will be changed, and anyone can go look for the very first example of it on wikipedia at this point to see what the argument is. We don't need to replay it in order for bystanders to know what's going to happen. So why bother playing it yourself. It's not even good practice for argument and debate. It dulls rather than sharpens the wit and mind.
I used to have a temptation to look down the conversational chess match and see the next 5 moves. If all of them will be taken exactly as predicted, there isn't much point in engaging further, and it's very easy to see where it starts from anymore that I just don't feel a need to be bothered with certain topics. People might learn they are making bad arguments that are easily and predictably swatted down and that following a certain playbook that everyone involved already knows isn't very useful as a result. But this is really unlikely. People involved will most likely disengage angrily, or at least dismissively with something like "well that's your opinion" or "agree to disagree".
That leaves the audience. And the audience is probably bored by the lack of movement/action or considers these too esoteric to bother with a firm opinion that would require investigation. Most people aren't libertarians. Most people aren't theologians or philosophers. Most people aren't gun policy advocates (one way or the other). Ergo, most people don't have much of a stake in the outcome.
Note: abortion isn't on the list. I think this is because there's more wiggle room. More people care at least a little about it, but don't know enough to feel they have a firm opinion. As such, people are often tempted to go for whatever and whoever has made the most compelling argument recently (hence we get a lot of ill-formed and poorly conceived anti-abortion legislation).
I'd also note one reason these debates aren't that interesting is that they're overrun by emotional reactions and stories rather than reasoned arguments.