On motivators to abandon classical logic
I'll touch first on your question, "[are there examples] of any concepts that cannot be formalized by current, known philosophical framework?" I'm tempted to simply say no; but I cannot prove that and won't try. What I would claim is that classical logic is robust enough that it can be extended with tools to surpass its limitations, and for this reason it is unlikely to be very fundamentally limited in its subject matter.
Quantum mechanics, for instance, is much beloved of people who would like to suggest that classical logic is flawed. For instance, the double-slit experiment suggests very strongly (and quite reasonably) that it is possible for something to be in two places at the same time — in a manner of speaking. By "a manner of speaking", I mean just that: it becomes important to determine what one means by "something" and "two places at the same time", if one is going to bother to broach the ontology of quantum mechanics, and whether there is "really really" something in two places at the same time will depend on that.
After mere confusion and bemusement, there are a few ways of reacting to this. One popular reaction is to conclude that classical logic must be wrong, because "logically" we known that something can only be in one place at a time. A more practical reaction is to note that classical logic doesn't contain any propositions about the locations of objects, and to conclude that perhaps "something can only be in one place at a time" is an empirically motivated axiom that may be put into doubt. One can ask whether a beam of light, or even a photon, is really a "thing", or if it is in a "place" if we aren't interacting with it. If you continue in such a way and want to avoid doing any mathematics, one might end up rediscovering the heavily antirealist (for microscopic particles, anyway) Copenhagen interpretation, which embraces classical logic but forbids you from actually positing any logical propositions which might keep you awake at night. It imposes such limitations by arriving at a heuristic for deciding when a proposition is "meaningless" according to simple epistemic criteria.
Garrett Birkhoff and John von Neumann proposed that, instead of formulating such heuristics to prevent you from entertaining a logical proposition or some logical conjunction of two logical propositions, you might formulate a system of quantum logic which would allow you to reason consistently about propositions; indeed, they suggested that perhaps the lesson of quantum mechanics is that the "effective" logic of reality at microscopic scales and smaller is not that logic which proved adequate until the 19th century. This is not a claim that classical logic is broken; but there is, as with all models and theories, a question of whether an alternative model for reasoning may be more useful in practise, just as one may more effectively solve certain analytic problems using integral calculus rather than working explicitly with infinite sums.
I can say that, even now, there is not yet a compelling argument for using quantum logic; although it has become important to recognise its potential significance as a source of insights when working on certain problems in quantum information theory (e.g. quantum generalizations of combinatorial probabilistic results). Even should we determine that it is better to describe reality using quantum logic rather than the classic boolean logic, this will not stop us from using boolean logic in everyday affairs, just as we still use Newtonian mechanics (and not, say, General Relativity) to build bridges.
Of course, if we were inclined to look for necessary conditions for giving up boolean logic, the classic Sorites paradox (how many grains of sand in a "pile" of sand?) would be a much better candidate than quantum mechanics, due to its accessibility. It is also illustrative of how it might be difficult to produce conditions necessitating the rejection of classical logic, because there is also a millenias-old practical approach to solving problems such as the Sorites: rather than asking yes/no questions about piles of sand, simply describe the shape, mass, and composition of the pile of sand — that is, describe it on a more precise level, using the tools of arithmetic and geometry, and restrict your use of boolean logic to reasoning in the mathematical theory instead of coming up with arbitrary and sophist-icated notions of "pileness". The difficulty in coming up with any phenomenon which cannot be solved using classical logic is that can always supply a sufficiently robust logical framework with more tools (arithmetic, geometry, calculus, category theory, material science...) to help it to do the job. The question is to what extent these extra tools are revealing the structure of reality, or instead amount to an analysis by epicycles, wherein the very complexity of the tool guarantees the existence of a solution but without any guarantees of simplicity or elegance.
If we take this to heart, there are likely no necessary conditions to abandon traditional logical frameworks; only sufficient conditions.
On non-motivators to abandon classical logic
The particular example you note of "unknown knowns", can be interpreted perfectly adequately using a traditional logical approach. From the introduction to the linked Wikipedia link to "unknown known":
There are at least two interpretations of Unknown knowns. The first is that they are things that we knew but have forgotten. The other is that they are the things that we know, but are unaware of knowing. The coining of the term is attributed to Slovenian Philosopher Slavoj Žižek and it refers to the unconscious beliefs and prejudices that determine how we perceive reality and intervene in it. It is the Freudian unconscious, the “knowledge which doesn’t know itself,” as Jacques Lacan said.
The label "unknown known" thus is a label which suggests a sort of gloss for two seemingly different concepts. Inasmuch as we have knowledge and world-models through rich associations to life and experience, these two concepts may point to different routes of investigating precisely the phenomenon described by "unconscious beliefs and prejudices that determine how we perceive reality".
However, the label "unknown known" is just a label, and one which is perhaps provocatively chosen; it does not refer in any meaningful way to any paradoxical structure in the concept to which it refers. A book which is stored on a library shelf but which is not catalogued is a similar sort of "unknown known", though library books do not quite play the same dynamical role, because researchers do not (often) go to the library and open up library books at random. In the case of the book, "unindexed content" would be an entirely uncontrovertial label for the same concept, referrring to the existence of data and metadata.
The human brain does not seem to have as clear a separation between data and metadata in the same way. What "unknown known" is trying to get at is how the residual network topology which remains from forgotten knowledge, or from knowledge which one has but which one applies reflexively / absent-mindedly / without awarenes, affects cognition. It can range from things which genuinely are knowledge (which one could remember, but forgets that one knows), to things which one could only remember with a mighty struggle and possibly re-constructing ideas or memories (i.e. artificially and in a process not unlike learning something anew). In this respect, even the label "knowledge" is inappropriate, in that it only very coarsely encapsulates the significance of the concept involved.
Finally, the idea that we should even concieve of it as "knowledge" is perhaps misguided. What is significant is not that it is knowledge which one might consider abstractly, but that it colours how one sees the world whether or not one even has access to it as information or an idea per se. So perhaps it might be better termed as a paraconception: not quite ideation, but related to it and taking place alongside it.
Inasmuch as anything that I've said above makes any sense, what we see is that classical logic does a perfectly good job of presenting a model for the specific example of an "unknown known", once one pushes past the mere label of the thing to consider the actual phenomenon which one wishes to discuss.
On the role of qualia in influencing logic
Regarding your edit, the things which you describe (unusual mental states, unusual sensory experiences, and avant-garde artistic works) don't really in themselves have any content upon which logic may act.
A novel or unusual experience may change elements of the inner structure or chemical balance of the brain, and thereby influence how things are perceived, but they do not consist themselves of propositions upon which logic may be brought to bear — and to be seen to fail. It is the conceptions — the way in which one fits one's experiences into a framework of experience in order to make sense of the world — with which one may attempt to reason logically, so the qualia in themselves cannot necessitate a change in one's logical mode of reasoning. They may motivate a change in how one theorizes about the world — the assumptions and hypotheses that one entertains, the questions that one regards as interesting — but this is different from changing what one considers to be the structure of a logical argument. Logic is just the way that one tries to combine ideas to extract the consequences of those ideas, in a way which preserves the truth of those ideas — and we have so far found that, in practise, one can change the ideas with which one reasons without changing the logic with which one tries to reason about them.
It is possible that an extrasensory experience, or a particularly profound "normal" sensory experience, might suggest ways of conceiving the world which are so radically different that it also suggests new ways to reason effectively, or that it demonstrates that some accepted ways of reasoning are actually not effective. This has arguably happened already, for reasons which are more mundane. Around the time that foundations of mathematics were being explored in the late 19th century, a schism of this sort arose concerning reasoning about infinities, and whether classical modes of reasoning such as reductio ad absurdum and double negation were valid ways of reasoning about the existence of individual elements within infinite collections. This gave rise a divergent notions of effective modes of reasoning, intuitionism/constructivism, which among other things only regards the existence of mathematical objects as established if a specific example can be exhibited by valid modes of construction (rather than, for instance, showing that a contradiction arises if one assumes that no object of the kind exists). Even though I generally accept "traditional" logic, I find this stronger standard of valid reasoning worthwhile; it presents a standard of reasoning which is aligned well with my own epistemic values, and forces one to reason in such a way that one has a more complete knowledge of the object under consideration. But it is also harder to work with, and classical logic does not seem to be any less reliable, so much as it opens up easier but less informative routes to the same conclusions which constructivism does.
So, as with the example of quantum mechanics above, how one attempts to conceive of the world actually can motivate a change in what forms of arguments — what logic — one accepts as generally valid. But it also does not necessarily do so; it's only a question as which mode of logical reasoning one finds most effective for a purpose. Indeed, it is possible to experiment with or embrace different modes of logical reasoning according to one's purpose — essentially selecting them according to some pre-logical and essentially aesthetic grounds, based on one's experience of what has proven effective in the past (which is not too different from empirical testing of theories in general).