A story should finish what it starts.
You control what, exactly, you choose to start. If you're not going to be finishing a murder mystery with a solution, you need to be careful not to set the story up in a way that the story will be unsatisfying without a solution.
Let's imagine you've got a mystery you don't want to solve. A few examples:
- The point is that the detective is obsessed, and the real climax is that he should just let it go.
- Or the point is that some mysteries are unsolvable, and we must all live with uncertainty.
- Or that the characters are wrong to be investigating this mystery; they've misunderstood everything and they're barking up the wrong tree.
Each of these is an example of a story, where the solution to the mystery is unimportant. That's why these stories probably shouldn't start out by establishing the mystery to be solved; instead, they should start out by establishing the problem that needs to reach resolution.
Don't begin with a dead body and questions to be answered; instead, consider:
- Beginning with the obsessed detective, showing how focused he is, and how oblivious (and destructive) he is during his pursuit of the case
- Beginning with the dread of uncertainty, showing how the characters are aching to find meaning and order in the world.
- If the characters are going to misunderstand what's important -- maybe start with what is important, and you'll be portraying your characters as being drawn away from that.
These are simply examples to illustrate the premise: your first, outermost story is what readers will be expecting you to resolve. That's what establishes what the story is about. That's the promise you're making.
Within that supporting framework, "inner" plot threads can already have resolutions like "the solution is, there is no solution." What makes that work is, these resolutions will be meaningful for the higher-level story arc. They'll be delivering on your bigger promise.
That being said, it's not clear to me that you want to resolve your mystery thread at all. You see it as a vehicle to get your characters moving; once it does that, it sounds like you don't see the benefit of actually resolving it at all.
To which I would say: a story should finish what it starts.
If you used a mystery to grab the reader's attention, you don't get to let that mystery drift off just because it's not useful to you any longer. The reader's attention is still on it. The reader is still expecting some form of payoff for it, and will feel cheated if they don't get it.
So: keep close track of what promises you're making the reader. Keep promises that you can; don't make promises that you can't. And if you feel like you need to make a promise that you can't really keep, that's a good sign that there's something important there that will have a payoff, and what you need to do is tweak the promise so it matches what you're actually going to provide.