Example of a $1$-categorical statement that really doesn't hold $\infty$-categorically In many introductions to $\infty$-category theory (in the sense of Lurie and Joyal, so quasicategories), it is explained that many (if not most) $1$-categorical statements have analogues in $\infty$-categories that are true, although the  proof is often (if not always) more complicated. 
I was wondering what kind of exceptions there were to this rule, if any :

Is there a $1$-categorical statement that has no known analogue in $\infty$-category theory which is conjectured/proved to be true ? Or better yet, is there a $1$-categorical statement where we have strong evidence that there could not be a valid $\infty$-analogue ? 

("strong evidence" could mean anything from counterexamples to the naive analogues to good heuristic reasons why an analogue shouldn't hold)
If there is no known example of this kind, this leads me to a second question : 

Is there any work on trying to automatize the process of going from $1$-CT to $\infty$-CT, e.g. a "translation theorem" that would say something along the lines of "if a statement $P$ [add technical hypotheses] holds in CT, then statement $f(P)$ holds in $\infty$-CT" where $f$ would be some explicit way of transforming a $1$-CT statement to an $\infty$-CT statement ? 

(of course, as always with quasicategories, $\infty$-category here actually means $(\infty,1)$-category)
 A: For a somewhat vague example, "There is a natural useful concept of elementary topos" has been having some trouble. Shulman has conjectured a definition but things are quite tricky, as the 1-categorical notion of elementary topos is finitary, and that's one of its key desiderata, but any sort of $\infty$-topos seems to inherently require "infinite" objects such as the loop space $\mathbb{Z}$ of the circle. It is also difficult to understand what a finitary notion of $\infty$-equivalence relation should be, as this is pretty well understood in Grothendieck toposes to be about the geometric realization of simplicial objects. All this apparent inherent infinitary behavior is a significant obstacle for homotopy type theory's efforts to, in one interpretation, give an internal language for $\infty$-toposes interpretable on a computer. 
A more concrete statement that is specifically known to be false: "Every Grothendieck topos is of the form 'sheaves on some site.'" The $\infty$-toposes which are sheaves on a site are, for instance, hypercomplete, a property with explicit counterexamples among general $\infty$-toposes. 
For a more elementary counterexample, the formula expressing any colimit as a coequalizer does not hold in an $\infty$-category. However, it is true that any $\infty$-category admitting coproducts and pushouts is cocomplete; one must, however, perform an infinite (there it is again!) iterative construction to build a general colimit from these. 
I don't really have anything to say on the closing question, except that I see no reason to think there should be such an automatic process. There are various efforts to express as much $\infty$-category theory in terms of ordinary category theory as possible, such as the program of Riehl and Verity and a part of the theory of derivators, but this isn't quite what you're asking.
