I’d like to expand on the accepted answer and comments.

Cisinski describes in §7.5.25 of his book how to compute right derived functors in the context of an $\infty$-category $C$ with weak equivalences and fibrations. This is good enough to compute $RF$ if $F:K(A):=N_{\mathrm{dg}}(\operatorname{Ch}A)\to D$ with $A$ a Grothendieck abelian category and $D$ any $\infty$-category: one takes quasi-isomorphisms as the weak equivalences in $K(A)$ and the fibrations to be as in the model structure on $\operatorname{Ch}A$ of HA.1.3.5.3. That this satisfies the axioms of Cisinski’s Definition 7.4.12 is easy to check, since a square in $K(A)$ is cartesian iff it’s cocartesian iff the map on cones is an equivalence in $K(A)$; i.e. a chain homotopy equivalence (HA.1.2.4.14 & HA.1.3.2.17). If $D=D(B):=K(B)[\mathrm{q.i.}^{-1}]$ with $B$ an abelian category then this computes $RF$ for any additive functor $F:A\to B$.

What about tensor product? Well, in that case one would like to find cofibrations that together with the weak equivalences make $K(A)$ into an $\infty$-category with weak equivalences and cofibrations and so that the cofibrations objects are the K-flat complexes. I don’t know how to do this in general, but when $A=\operatorname{Mod}_R$ then a preprint of Gillespie shows there exists a model structure on $\operatorname{Ch}A$ whose cofibrant objects are the K-flat complexes. In any case, it seems at least superficially easier to compute with Deligne’s formula than to find the right fibrations or cofibrations, so let’s turn to that.

Proposition Let $C$ and $W$ be as in §7.2.1 of Cisinski’s book and let $D$ be a (co)complete $\infty$-category. Suppose given a right (left) calculus of fractions $W(x)$ at $x\in C$ and let $\gamma:C\to C[W^{-1}]$ be the localization map. Then $$(\gamma_*F)(\gamma x)\simeq\lim_{z_0\to x_0}F(z),$$ where the limit is indexed over $W(x)$ (respectively $$(\gamma_!F)(\gamma x)\simeq\operatorname{colim}_{x_0\to z_0}F(z),$$ indexed over $W(x)$).

(Here, $\gamma_*:\operatorname{Fun}(C,D)\to\operatorname{Fun}(C[W^{-1}],D)$ is the functor of right Kan extension, so $\gamma_*F$ computes $LF$, while $\gamma_!:\operatorname{Fun}(C,D)\to\operatorname{Fun}(C[W^{-1}],D)$ is left Kan extension and $\gamma_!F=RF$.)

Corollary Let $A$ be an abelian category, $D$ a cocomplete $\infty$-category, and $F:K(A)\to D$ a functor. If $K\in K(A)$, $$RF(K)\simeq\operatorname{colim}_{K\to K’} F(K’),$$ where the (filtered) colimit is indexed by $K(A)_{K/}^{\mathrm{q.i.}}$, the full subcategory of $K(A)_{K/}$ on the quasi-isomorphisms. Dually, if $D$ is instead complete, $$LF(K)\simeq\lim_{K’\to K}F(K’),$$ where the (cofiltered) limit is indexed by $K(A)_{/K}^{\mathrm{q.i.}}$.

The corollary follows from the proposition using Cisinski’s Theorem 7.2.16, as the set of quasi-isomorphisms is closed under composition as well as pullback and pushout in $K(A)$.

Proof of Proposition – We can rewrite (the dual of) Cisinski’s Corollary 7.29 as follows: if $F$ is a functor $C\to\mathcal S(=\mathrm{Kan})$ and there is a right calculus of fractions $W(x)$ at $x\in C$, then $$\lim_{z_0\to x_0}F(z)\simeq(\gamma_*F)(\gamma x),$$ where the limit is indexed over $W(x)$. If $G:X\to Y$ is any functor of simplicial sets and $x\in X$, let $G_x:=x^*G$ denote the $x$-fiber of $G$. The functor $$\gamma_*:=\operatorname{Fun}(X,\gamma_*):\operatorname{Fun}(X,\operatorname{Fun}(C,\mathcal S))\to\operatorname{Fun}(X,\operatorname{Fun}(C[W^{-1}],\mathcal S))$$ is functorial in $X$, so in particular $\gamma_*(\Phi)_x=\gamma_*\Phi_x$ for any $x\in X$ and functor $\Phi:X\to\operatorname{Fun}(C,\mathcal S)$. Letting $X=D^{\mathrm{op}}$ and $\Phi=h_DF=\operatorname{Map}_D(-,F):C\to\operatorname{Fun}(D^{\mathrm{op}},\mathcal S)$, we produce via right Kan extension a functor $\gamma_*(h_DF):C[W^{-1}]\to\operatorname{Fun}(D^{\mathrm{op}},\mathcal S)$ so that for each $d\in D$ and $x\in C$, $$(\gamma_*(h_DF)_{\gamma x})_d=\gamma_*\operatorname{Map}_D(d,F)_{\gamma x}=\lim_{z_0\to x_0}\operatorname{Map}_D(d,Fz)=\operatorname{Map}_D(d,\lim_{z_0\to x_0}F(z)).$$ On the other hand, the functor $$\alpha:\operatorname{Fun}(C[W^{-1}],h_D):\operatorname{Fun}(C[W^{-1}],D)\to\operatorname{Fun}(C[W^{-1}],\operatorname{Fun}(D^{\mathrm{op}},\mathcal S))$$ sends $\gamma_*F=LF$ to $\gamma_*(h_DF)$ by Cisinski’s Proposition 6.4.9. Therefore $$(\gamma_*(h_DF)_{\gamma x})_d=(\alpha(LF)_{\gamma x})_d=\operatorname{Map}_D(-,LF(\gamma x))_d=\operatorname{Map}_D(d,LF(\gamma x)).\qquad\square$$

I’d like to expand on the accepted answer and comments.

Cisinski describes in §7.5.25 of his book how to compute right derived functors in the context of an $\infty$-category $C$ with weak equivalences and fibrations. This is good enough to compute $RF$ if $F:K(A):=N_{\mathrm{dg}}(\operatorname{Ch}A)\to D$ with $A$ a Grothendieck abelian category and $D$ any $\infty$-category: one takes quasi-isomorphisms as the weak equivalences in $K(A)$ and the fibrations to be as in the model structure on $\operatorname{Ch}A$ of HA.1.3.5.3. That this satisfies the axioms of Cisinski’s Definition 7.4.12 is easy to check, since a square in $K(A)$ is cartesian iff it’s cocartesian iff the map on cones is an equivalence in $K(A)$; i.e. a chain homotopy equivalence (HA.1.2.4.14 & HA.1.3.2.17). If $D=D(B):=K(B)[\mathrm{q.i.}^{-1}]$ with $B$ an abelian category then this computes $RF$ for any additive functor $F:A\to B$.

What about tensor product? Well, in that case one would like to find cofibrations that together with the weak equivalences make $K(A)$ into an $\infty$-category with weak equivalences and cofibrations and so that the cofibrant objects are the K-flat complexes. I don’t know how to do this in general, but when $A=\operatorname{Mod}_R$ then a preprint of Gillespie shows there exists a model structure on $\operatorname{Ch}A$ whose cofibrant objects are the K-flat complexes. In any case, it seems at least superficially easier to compute with Deligne’s formula than to find the right fibrations or cofibrations, so let’s turn to that.

Proposition Let $C$ and $W$ be as in §7.2.1 of Cisinski’s book and let $D$ be a (co)complete $\infty$-category. Suppose given a right (left) calculus of fractions $W(x)$ at $x\in C$ and let $\gamma:C\to C[W^{-1}]$ be the localization map. Then $$(\gamma_*F)(\gamma x)\simeq\lim_{z_0\to x_0}F(z),$$ where the limit is indexed over $W(x)$ (respectively $$(\gamma_!F)(\gamma x)\simeq\operatorname{colim}_{x_0\to z_0}F(z),$$ indexed over $W(x)$).

(Here, $\gamma_*:\operatorname{Fun}(C,D)\to\operatorname{Fun}(C[W^{-1}],D)$ is the functor of right Kan extension, so $\gamma_*F$ computes $LF$, while $\gamma_!:\operatorname{Fun}(C,D)\to\operatorname{Fun}(C[W^{-1}],D)$ is left Kan extension and $\gamma_!F=RF$.)

Corollary Let $A$ be an abelian category, $D$ a cocomplete $\infty$-category, and $F:K(A)\to D$ a functor. If $K\in K(A)$, $$RF(K)\simeq\operatorname{colim}_{K\to K’} F(K’),$$ where the (filtered) colimit is indexed by $K(A)_{K/}^{\mathrm{q.i.}}$, the full subcategory of $K(A)_{K/}$ on the quasi-isomorphisms. Dually, if $D$ is instead complete, $$LF(K)\simeq\lim_{K’\to K}F(K’),$$ where the (cofiltered) limit is indexed by $K(A)_{/K}^{\mathrm{q.i.}}$.

The corollary follows from the proposition using Cisinski’s Theorem 7.2.16, as the set of quasi-isomorphisms is closed under composition as well as pullback and pushout in $K(A)$.

Proof of Proposition – We can rewrite (the dual of) Cisinski’s Corollary 7.29 as follows: if $F$ is a functor $C\to\mathcal S(=\mathrm{Kan})$ and there is a right calculus of fractions $W(x)$ at $x\in C$, then $$\lim_{z_0\to x_0}F(z)\simeq(\gamma_*F)(\gamma x),$$ where the limit is indexed over $W(x)$. If $G:X\to Y$ is any functor of simplicial sets and $x\in X$, let $G_x:=x^*G$ denote the $x$-fiber of $G$. The functor $$\gamma_*:=\operatorname{Fun}(X,\gamma_*):\operatorname{Fun}(X,\operatorname{Fun}(C,\mathcal S))\to\operatorname{Fun}(X,\operatorname{Fun}(C[W^{-1}],\mathcal S))$$ is functorial in $X$, so in particular $\gamma_*(\Phi)_x=\gamma_*\Phi_x$ for any $x\in X$ and functor $\Phi:X\to\operatorname{Fun}(C,\mathcal S)$. Letting $X=D^{\mathrm{op}}$ and $\Phi=h_DF=\operatorname{Map}_D(-,F):C\to\operatorname{Fun}(D^{\mathrm{op}},\mathcal S)$, we produce via right Kan extension a functor $\gamma_*(h_DF):C[W^{-1}]\to\operatorname{Fun}(D^{\mathrm{op}},\mathcal S)$ so that for each $d\in D$ and $x\in C$, $$(\gamma_*(h_DF)_{\gamma x})_d=\gamma_*\operatorname{Map}_D(d,F)_{\gamma x}=\lim_{z_0\to x_0}\operatorname{Map}_D(d,Fz)=\operatorname{Map}_D(d,\lim_{z_0\to x_0}F(z)).$$ On the other hand, the functor $$\alpha:\operatorname{Fun}(C[W^{-1}],h_D):\operatorname{Fun}(C[W^{-1}],D)\to\operatorname{Fun}(C[W^{-1}],\operatorname{Fun}(D^{\mathrm{op}},\mathcal S))$$ sends $\gamma_*F=LF$ to $\gamma_*(h_DF)$ by Cisinski’s Proposition 6.4.9. Therefore $$(\gamma_*(h_DF)_{\gamma x})_d=(\alpha(LF)_{\gamma x})_d=\operatorname{Map}_D(-,LF(\gamma x))_d=\operatorname{Map}_D(d,LF(\gamma x)).\qquad\square$$