Is there a functor that preserves all small limits but not a large one?
Let's try this. I'll use colimits, so take the opposite.
The class of all ordinals is ordered. Add one more element $\infty$ at the end, bigger than all of them. View this "large ordered set" as a large category $\cal C$. A small diagram in $\cal C$ has colimit $\infty$ if $\infty$ occurs in the diagram, and otherwise it has a colimit less than $\infty$. But the large diagram consisting of everything except $\infty$ has colimit $\infty$. The functor to the ordered set $\lbrace 0<1\rbrace$ that sends $\infty$ to $1$ and everything else to $0$ preserves small colimits but not all colimits.