What is a model category from an $\infty$ point of view? A number of different models for $\infty$ categories can seen to have analogs in $\infty$-category theory. For example:


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*Quasicategories: $\Delta \subseteq \mathrm{Cat}_{(\infty, 1)}$ is (?) a dense generator, so we can see $\infty$-categories as presheaves on $\Delta$

*Simplicially enriched categories: Simplicial enrichment is a proxy for enrichment in $\mathrm{Cat}_{(\infty, 0)}$

*Simplicial categories: These can be seen as the objects in an intermediate step towards adjoining simplicially-indexed colimits to $\mathrm{Cat}_{(1,1)}$

*Complete segal spaces: These look to be inspired by models in $\mathrm{Cat}_{(\infty, 0)}$ of the finite limit sketch defining categories

*Relative categories: A pair $(C,W)$ is a proxy for the localization $C[W^{-1}]$.


However, I don't know of a suitable interpretation of a model structure.
Every model category is in particular a relative category, so that's how they correspond to $\infty$-categories. But that doesn't involve the fibrations and cofibrations.
My question is if there is some notion in $\infty$-category theory that is analogous to equipping a relative category with a choice of classes of fibrations and cofibrations.
 A: This is just a long comment, which does not really address the main question.
Instead, I give few examples where fibrations are an additional rigidity structure allowing "homotopical commutativity".
In all of the simplicial models for $(\infty, 1)$-categories that you listed, only the fibrant objects of the respective model category structure behave as "categories weakly enriched in $\infty$-groupoids".
Inside this categorical framework (if you want, formal 2-categorical), there is a notion of localisation $C[W^{-1}]$ of an $\infty$-category $C$ with respect to a sub-$\infty$-category $W$. Given a relative category $(C, W)$, it is a theorem that the $\infty$-localisation $N(C)[N(W)^{-1}]$ models the $\infty$-category associated to $(C, W)$ (see for instance the appendix of Brown categories and bicategories of Geoffroy Horel).
However if $(C, W)$ is not fibrant, nor will be its Rezk's relative nerve, i.e. $N^R(C, W)$ is not really an $\infty$-category.
Suppose now your relative category $(C, W)$ has an additional structure of fibrantions that interacts gently with the weak equivalences (for instance, a model category), let's call such a structure a fibration category. Lennart Meier showed in Fibration categories are fibrant relative categories that any such relative category is fibrant in the Barwick-Kan model category structure and thus $N^R(C, W)$ is an $\infty$-category modelling the $\infty$-categorical localisation $N(C)[N(W)^{-1}]$.
This is just the reflection of something happening entirely in the $\infty$-categorical world. Indeed, in Chapter 7 of Higher categories and homotopical algebra Denis-Charles Cisinski studies the $\infty$-localisation theory of fibration $\infty$-categories.
Within this framework, we have the beautiful feature that the $\infty$-localisation of a fibration $\infty$-category has finite $\infty$-limits (see Section 7.5).
Moreover the identification of homotopy limits and $\infty$-limits can be stated as an elegant theorem about an equivalence of $\infty$-categories (see Theorem 7.9.8 and Remark 7.9.10).
The longer story tells that fibration $\infty$-categories are closed by exponentiation, that is if $(C, W)$ is a fibration $\infty$-category and $I$ is an $\infty$-category, then $\underline{\text{Hom}}(I, C)$ is still a fibration $\infty$-category, with fiber-wise weak equivalences $W_I$. If $(C, W)$ is just (the nerve of) a regular fibration category and $I$ (the nerve of) a small category, then the $\infty$-localisation $\text{Fun}(I, C)[W_I^{-1}]$ is equivalent to the $\infty$-functor category $\underline{\text{Hom}}(I, C[W^{-1}])$. (Notice that $(\text{Fun}(I, C), W_I)$ is a fibrant relative category!)
In the case where the fibration $\infty$-category is actually a model $\infty$-category, the theory has been studied by Aaron Mazel-Gee, as Dmitri Pavlov says in his nice answer.
A: One can make a reasonable claim that the analog of model categories in the realm of ∞-categories are model ∞-categories,
see, for instance, http://arxiv.org/abs/1412.8411 and other papers by Aaron Mazel-Gee.
Relative categories encode an ∞-categories,
whereas the additional data of a model structure can be seen
as encoding (say) choices of limit-colimit diagrams for which limits and colimits commute, or (more generally) choices of left/right adjoint functors that preserve some specific limits/colimits.
Different choices of model structures produce different
commuting limits/colimits.
This is explained in more detail here:
Why do we need model categories?
In the world of ∞-categories one still needs to commute ∞-limits and ∞-colimits
that need not commute in general,
or commute an ∞-limit past a left adjoint functor, etc.
This is what model ∞-structures accomplish.
Thus model ∞-categories perform the same role in the realm of ∞-categories
as model categories do in the realm of 1-categories.
