As many have pointed out by now, model theory, as usually practiced nowadays, doesn't require a lot of attention to foundations; the answers to questions in tame model theory like neostability theory are usually absolute. But I wanted to mention a few examples where moving beyond ZFC was important to bring core model-theoretic phenomena to light.
Shelah's 'dividing lines' philosophy, which has had a huge impact on the development of model theory, suggests that it is interesting to find properties that divide the space of first order theories into tame and wild. Good dividing lines have the property that you can prove interesting structure theorems on the tame side and interesting nonstructure theorems on the wild side. But they also have the property that they can be characterized by both 'inside' (i.e. syntactic) and 'outside' (i.e. semantic or set-theoretic) criteria. The outside characterization usually has a set-theoretic flavor and this sometimes requires working outside of ZFC to obtain the equivalence. Here are some examples:
1. NIP - A theory $T$ is NIP if no formula has the independence property, i.e. there is no $\varphi(x;y)$ and sequence $(a_{i})_{i \in \mathbb{N}}$ in a model of $T$ such that $\{\varphi(x;a_{i}) : i \in X\} \cup \{\neg \varphi(x;a_{i}) : i \not\in X\}$ is consistent for all $X \subseteq \mathbb{N}$. This is equivalent to an outside condition: define $\mathrm{ded}(\kappa)$ to be the supremum of cardinals $\lambda$ such that there is a linear order of size $\lambda$ with a dense subset of size $\kappa$. Shelah proved that the number of types over a set of size $\kappa$ in a countable NIP theory can be bounded above by the cardinal $\mathrm{ded}(\kappa)^{\aleph_{0}}$, while the number of types over a set of size $\kappa$ in any theory with the independence property will be $2^{\kappa}$. So one can characterize the (countable) NIP theories by their type counting function, provided $\mathrm{ded}(\kappa )^{\aleph_{0}} < 2^{\kappa}$ for some $\kappa$. This can be arranged by forcing results of Mitchell.
2. Simple theories - A theory $T$ is simple if no formula has the tree property, i.e. there is no $\varphi(x;y)$, tree of tuples $(a_{\eta})_{\eta \in \omega^{<\omega}}$ in a model of $T$, and $k < \omega$ such that
(a) For all $\eta \in \omega^{\omega}$, $\{\varphi(x;a_{\eta | i} : i < \omega\}$ is consistent.
(b) For all $\eta \in \omega^{<\omega}$, $\{\varphi(x;a_{\eta^{\frown}\langle i \rangle}) : i < \omega\}$ is $k$-inconsistent.
This is equivalent to an outside condition. Define the saturation spectrum $\mathrm{SP}(T)$ to be the set of pairs of cardinals $\kappa \leq \lambda$ such that every model of $T$ of size at most $\lambda$ has a $\kappa$-saturated elementary extension of size $\lambda$. It follows from a standard argument that if $\lambda = \lambda^{<\kappa}$, then $(\kappa,\lambda)$ is in the saturation spectrum of any $T$ (in a countable language, say) and Shelah proves that, modulo a forcing axiom he calls $\mathrm{Ax}_{0}\mu$, a theory $T$ is simple if and only if there is some pair $(\lambda, \kappa)$ in the saturation spectrum of $T$ which does not satisfy $\lambda = \lambda^{<\kappa}$. Shelah proves the consistency of this forcing axiom by a class forcing argument.
3. NSOP$_{2}$ A theory $T$ is said to be NSOP$_{2}$ if there is no formula $\varphi(x;y)$ and tree of tuples $(a_{\eta})_{\eta \in \omega^{<\omega}}$ in a model of $T$ such that
(a) For all $\eta \in \omega^{\omega}$, $\{\varphi(x;a_{\eta | i}) : i < \omega\}$ is consistent.
(b) For all $\eta \perp \nu$ in $\omega^{<\omega}$, $\{\varphi(x;a_{\eta}), \varphi(x;a_{\nu})\}$ is inconsistent.
This has an outside characterization as well, via something called the interpretability order (sometimes also called the `triangle star order'). Suppose $T_{1},T_{2}$ are countable theories and $\lambda$ is a cardinal. Say $T_{1} \unlhd^{*}_{\lambda} T_{2}$ if there is a theory $\tilde{T}$ in a language of size $< \lambda$ that interprets both $T_{1}$ and $T_{2}$ and, moreover, has the property that, in any model of $\tilde{T}$, if the interpreted model of $T_{2}$ is $\lambda$-saturated, then the interpreted model of $T_{1}$ is $\lambda$-saturated. Then say $T_{1} \unlhd^{*} T_{2}$ if $T_{1} \unlhd^{*}_{\lambda} T_{2}$ for all sufficiently large $\lambda$.
Assuming GCH, a countable theory is maximal in this pre-order $\unlhd^{*}$ on theories if and only if that theory has SOP$_{2}$. This is really three theorems: Džamonja-Shelah proved that maximality implies a property called SOP$_{2}''$ assuming GCH. Shelah-Usvyatsov proved that SOP$_{2}''$ and SOP$_{2}$ are equivalent for theories, and then Malliaris-Shelah proved recently that SOP$_{2}$ implies maximality.
The point of these examples is that set-theoretic characterizations have been established as part of the method for recognizing dividing lines in model theory and often these characterizations require making assumptions that go beyond what can be established in ZFC alone. It is no doubt true that most questions about simple or NIP theories are absolute, but the fact that they can be given this sort of abstract set-theoretic description is part of what contributes to the sense that they mark meaningful notions of complexity.
