Does leaving "any" change the meaning of the sentence?

As an example to illustrate why there is a difference between the two, let's switch the word "human" with "woman" (bear with me on this...)

It tackles a problem with no preconception that any woman would have.

By saying "any woman" it suggests that you have some kind of knowledge and insight into all the different preconceptions that women may have. However, the reader or listener may assume that it does not exclude preconceptions held by men.

Likewise, in your original sentence, "any human" suggests an understanding of human preconceptions and it authoritatively assures that all have been taken into consideration. It does leave me wondering why it says "any human" and not "anyone". Is the intention that dogs and cats have preconceptions too, but that you haven't excluded these?

It tackles a problem with no preconception that women would have.

Hopefully, this is where you see the sense of my example - because by using "women" this version of the sentence sounds kinda sexist. It feels like a sweeping generalisation that all women have the same preconceptions. I'd find it quite offensive.

Likewise, with "humans", it suggests that we all have the same preconceptions when that is patently untrue.

As a native speaker, I prefer the first example over the second for the reasons I give - it seems less like a sweeping generalisation about preconceptions. However, if the scope of your statement is entirely limited to human beings then I would prefer "anyone" to remove the possibility that you are considering preconceptions of other creatures too.

any human = a single human (out of all possible humans)

humans = humans (in general)

The second is referring more generally to humans as a collective group, whereas the former makes explicitly clear that we could scrutinize every single individual human and not find even one fitting the criteria.

Note that this use of any is restricted to non-affirmative contexts.