This seems difficult to answer in any kind of uniform way, i.e., it seems we have to work on a case-by-case basis. Let's start with $\mathrm{Top}_{\mathrm{open}}$, the category of topological spaces and open continuous maps.
A useful heuristic is to let oneself be guided by locales instead of topological spaces. (For general information on locales, see for instance Mac Lane-Moerdijk, Sheaves in Geometry and Logic, chapter IX.) Recall that the category of locales is (by definition) the category opposite to the category of frames (sup-lattices in which finite meets distribute over arbitrary joins); the frame associated with a locale (by viewing the same object in the opposite category) should be thought of as its topology. Now a frame automatically carries Heyting algebra structure, and it turns out that open continuous maps of locales correspond to frame maps that are also Heyting algebra maps (i.e., Heyting algebra maps which are sup-preserving). Thus, the category of locales and open continuous maps is opposite to the category of cocomplete Heyting algebras, which is a category of algebras of an infinitary algebraic theory, aka a category monadic over $Set$, and all such monadic categories are complete and cocomplete. It follows that the category of locales and open continuous maps is cocomplete and complete. Moreover, limits of cocomplete Heyting algebras are computed just as they are in $Set$, and so the calculation of colimits in locales and open continuous maps will be similarly straightforward.
Guided by this heuristic, one might at the very least expect that colimits in $\mathrm{Top}_{\mathrm{open}}$ are calculated just as they are in $\mathrm{Top}$. Indeed, one can easily check this directly for arbitrary coproducts (where the coproduct topology on a disjoint union $X = \sum_\alpha X_\alpha$ is the largest one such that all the canonical inclusions $X_\alpha \to X$ are continuous; notice they are open). As for coequalizers, if two maps
$$f, g: R \stackrel{\to}{\to} X$$
are open and continuous, and if we consider the coequalizer $q: X \to Y$ in $\mathrm{Top}$ (where the topology on $Y$ is the usual quotient topology induced by $q$), then we claim that $q$ is also open and this is the coequalizer in the category of open continuous maps. For, if $V$ is open in $X$, then $q(V)$ is open in $Y$ iff $q^{-1}(q(V))$ is open in $X$, and clearly this holds since
$$q^{-1}(q(V)) = V \cup \bigcup_n (g f^{-1})^n(V) \cup (f g^{-1})^n(V)$$
where the right side is open because $f$ and $g$ are open and continuous.
Pausing briefly to consider the category of topological spaces and closed continuous maps, it is certainly true that finite coproducts exist and are computed just as they are in $\mathrm{Top}$. Also, it is true that quotients of equivalence relations are computed as in $\mathrm{Top}$. That is to say: if $E \subseteq X \times X$ is an equivalence relation and the two projection maps $\pi_1: E \to X$ and $\pi_2: E \to X$ are closed, then the coequalizer $q = \mathrm{coeq}(\pi_1, \pi_2)$ in $Top$ is the coequalizer in the category of closed continuous maps. Indeed, if $C$ is closed in $X$, then $q(C)$ is closed because
$$q^{-1}(q(C)) = \pi_1(\pi_2^{-1}(C))$$
is closed.
Limits are trickier. Let's again take $\mathrm{Top}_{\mathrm{open}}$. One should not expect to calculate limits as one would in $\mathrm{Top}$ (for example, Cantor space as a countable product of copies of the discrete two-element space could not possibly be the correct product in $\mathrm{Top}_{\mathrm{open}}$ since Cantor space has no isolated points), nor even as in $\mathrm{Set}$, so this makes limits a little weird. However, if we have a diagram $F: D \to \mathrm{Top}_{\mathrm{open}}$ and all the spaces $F(d)$ are sober (i.e., of the form $\mathrm{Pt}(\mathcal{O}(X))$, where $\mathrm{Pt}: \mathrm{Frame}^{op} \to \mathrm{Top}$ is right adjoint to $\mathcal{O}: Top \to \mathrm{Frame}^{op}$), then the limit of $F$ in $\mathrm{Top}_{\mathrm{open}}$ can be constructed as $\mathrm{Pt}(\mathrm{colim} \; \mathcal{O}(F d))$ where the colimit is computed in the category of cocomplete Heyting algebras. This uses the fact that for any topological space $X$, the set of open continuous maps $X \to \mathrm{Pt}(H)$ ($H$ a cocomplete Heyting algebra) is in natural one-one correspondence with cocomplete Heyting algebra maps $H \to \mathcal{O}(X)$, together with elementary categorical manipulations.
I am unsure whether more general limits exist in $\mathrm{Top}_{\mathrm{open}}$.