I'm curious about people's thoughts on puzzles that don't have a solution. I've seen a couple of these posted in the last few weeks (since I've been on the forum) and they tend to generate a lot of conversation and views but also seem to be a bit unfair as they ask the puzzler for a solution that doesn't exist.
This is naturally going to be pure opinion, but I've got no problem with unsolvable problems so long as the puzzle takes the form: "Find the solution, or prove one doesn't exist". As @Brandon_J says in the comments, "proving that a puzzle is strictly unsolvable is often an excellent puzzle itself".
I think presenting something as having a solution, whilst it is actually impossible is disingenuous and falls into the realm of trickery rather than actually being a puzzle. Sometimes a poster may not know if the puzzle has a solution or not (e.g. some of the examples posted in comments on the question), and in this case I think there should generally be an assumed "or prove a solution doesn't exist" applied, though ideally this would be explicitly included.
Some additional thoughts on these types of puzzles.
I see these fall into one of these categories:
Puzzles posted as a challenge to, as Alconja put it, "Find the solution, or prove one doesn't exist"—we see these from time to time and (as @Brandon_J commented) "proving that a puzzle is strictly unsolvable is often an excellent puzzle itself."
When the puzzle directly asks if it is solvable, and for proof if not, there can be no question of fairness.
Puzzles that are known to be unsolved, posted here to attract interest or challenge people to see if they can provide any new thoughts that lead forward to a solution, and usually posted with (links to) relevant background and current results and theories. Occasionally these are posted by someone who doesn't know what they're asking is a known open question, but it's pretty common for someone else to recognize such a question for what it is, so any perceived unfairness to such questions is usually limited and temporary. The 100 soldier problem seems to belong to this category.
We have a tag unsolved-mysteries that fits most such questions.
Questions asked by someone frustrated by a puzzle they've encountered, on which they can't make any progress. These are usually posted as a question about the puzzle (as in, "I'm stuck, how does one solve a puzzle like this?"). The answer to such questions may well be "You can't, that's not solvable, because "—the Help solving Sliding Tiles Puzzle 2x3 question you reference, for example, turns out to be this.
Many such questions are at most minor variations of situations known to be unsolvable (often quickly demonstrated to be impossible for reasons of parity).
Asking for help with such a question isn't "unfair" at all—we welcome questions about Puzzles and their creation and solving, not just puzzle challenges, and someone legitimately asking why they're stuck is well within the charter of this site.
Puzzles which, due to how they're constructed, are "unsolvable" in that there's no answer that is clearly correct. Almost every IQ test question we get here, as well as things like Who is the Killer?, have intended answers (we hope!) but are so ambiguously designed that an answer other than the "official" one seems not only easily defensible, but is often actually better (frequently because the "official" answer relies on the kind of trickery Alconja mentioned and which puzzle setters would do well to eschew). For some of these questions, most if not all of the possible solutions can be convincingly argued for!
These are flawed puzzles, and we have a custom close reason for them.
Puzzles (re-)posted as a challenge which are undisclosed examples of category 2, 3 or 4. Here I'll weigh in and say that posting these when you know (or should know) they're unanswerable is indeed unfair. This would fit a pattern OP asks about, of intentionally trying to "generate a lot of conversation and views" (and, often, upvotes via HNQ drive-bys), "but also seem to be a bit unfair as they ask the puzzler for a solution that doesn't exist" which is exactly why they're effective at generating conversation and debate. However, despite OP's casual use of the word, we are not a forum in the usual sense, and generating conversation and debate is expressly not the purpose of the Q&A format here. Such questions are not only unfair but are often misuse of the platform.
Unfortunately it's sometimes hard to tell a 2, 3, or 4 apart from a 5. The best we can do is to assume good faith and provide the appropriate feedback on the question:
1s are, of course, just fine. Answer them if you can!
2s, if not disclosed as such, should be edited to include relevant detail about the open problem and its applicability to the question at hand.
3s are often duplicates; close as such if possible. If not, an answer explaining why they're unsolvable suffices.
4s should be closed.
5s, if habitually posted by a user, may warrant moderator intervention; flag if you see a pattern by one user.