Here is a reformulation/generalisation of G. Stefanich's counterexample, showing that sheaf-locality can fail very dramatically once we leave the realm of locally finitely presentable categories. More precisely:

Proposition.

1. There is an algebraic category $\mathcal{A}$ that (simultaneously) fails to be sheaf-local on every locally connected Hausdorff space $X$ with at least one open subset that is not closed. ($\mathcal{A}$ does not depend on $X$.)

2. For every uncountable regular cardinal $\kappa$, there is a $\kappa$-ary algebraic category $\mathcal{A}$ that (simultaneously) fails to be sheaf-local on every locally connected Hausdorff space $X$ of cardinality $< \kappa$ with at least one subset that is not closed. ($\mathcal{A}$ depends on $\kappa$ but not $X$.)

Proof. Recall that the Sierpiński space $S$ is the topological space whose set of points is $\{ 0, 1 \}$ and whose open subsets are $\emptyset, \{ 1 \}, \{ 0, 1 \}$. Almost by construction, there is a natural bijection between continuous maps $X \to S$ and open subsets of $X$, so we have a sheaf $\Omega$ on $X$ whose sections over an open $U \subseteq X$ are open subsets of $U$.

Similarly, there is a natural bijection between continuous maps $X \to 2$ and clopen subsets of $X$, so we have a sheaf $Q$ on $X$ whose sections over an open $U \subseteq X$ are the clopen subsets of $U$. Furthermore, the continuous injective map $2 \to S$ gives rise to a subsheaf inclusion $Q \to \Omega$.

Now let $\mathcal{A}$ be the category of $\kappa$-ary join semilattices, i.e. posets in which every subset of cardinality $< \kappa$ has a least upper bound, and assume $\kappa$ is greater than the cardinality of $X$. This is a $\kappa$-ary algebraic category (hence locally $\kappa$-presentable a fortiori). Although $\mathcal{A}$ has filtered colimits, they are not preserved by the forgetful functor $\mathcal{A} \to \textbf{Set}$ if $\kappa > \aleph_0$. This is the root cause of the failure of sheaf-locality for $\mathcal{A}$.

Let $N (x)$ be the filter of open neighbourhoods of $x$ in $X$. There is a natural morphism $\Omega (U) \to \Omega (1)$ in $\mathcal{A}$ sending $V$ to $1$ if and only if $x \in V$, so we get a morphism $\varinjlim_{U : N (x)^\textrm{op}} \Omega (U) \to \Omega (1)$. If $X$ is Hausdorff, this is an isomorphism: indeed, if $V \in \Omega (X)$ and $x \notin V$, then $V$ is the union of $< \kappa$ open $V_\alpha \subseteq V$ such that $x \notin \overline{V_\alpha} \subseteq X$; but if $V' \in \Omega (X)$ and $x \notin \overline{V'}$, then we can find $U \in N (x)$ with $U \cap V' = \emptyset$, so $V'$ must be identified with $\emptyset$ in the colimit, and hence $V$ is also identified with $\emptyset$ in the colimit. (This argument would also work when we consider $\Omega$ as a sheaf of sets if $N (x)^\textrm{op}$ is $\kappa$-filtered.)

On the other hand, if $X$ is locally connected, then it is easy to see that $\varinjlim_{U : N (x)^\textrm{op}} Q (U) \cong Q (1)$, being a colimit of a cofinally constant diagram. Thus, if $X$ is a locally connected Hausdorff space with at least one open subset that is not closed, then $Q_x \to \Omega_x$ is an isomorphism for every $x \in X$, but $Q \to \Omega$ is not an isomorphism.

The above proves the second claim of the proposition. For the first claim, remove the cardinality restrictions and replace $\mathcal{A}$ with the category of complete join semilattices.　◼