Let $\tau$ be a Grothendieck topology on the category of schemes (of suitably bounded cardinality if $\tau$ is ‘large’), $X$ a scheme, $\Lambda$ a ring, $\operatorname{Mod}_\Lambda$ the abelian category of $\Lambda$-sheaves on $X_\tau$, $\operatorname{Ch}:=\operatorname{Ch}^{\geq0}(\operatorname{Mod}_\Lambda)$ the abelian category of non-negative cochain complexes, $\mathcal D^{\geq0}(X)$ the derived $\infty$-category, $\mathcal F\in\mathcal D^{\geq0}(X)$ a sheaf (i.e. complex of sheaves). If $U\to X$ is a map of schemes, let $R\Gamma(U_\tau,-):\mathcal D^{\geq0}(X)\to\mathcal D^{\geq0}(\Lambda)$ be the derived functor of global sections of the pullback of $\mathcal F$ (defined via HA.1.3.3.2 or equivalently [as discussed here][1] as a (homotopy) Kan extension). Then $R\Gamma(-,\mathcal F)$ takes finite disjoint unions to finite products and satisfies the $\infty$-categorical sheaf axiom that you state; i.e. for every covering map $f:U\to X$ in the topology $\tau$ we have
$$R\Gamma(X_\tau,\mathcal F)=\lim\Big(R\Gamma(U,\mathcal F)\rightrightarrows R\Gamma(U\times_XU,\mathcal F)\mathrel{\substack{\textstyle\rightarrow\\[-0.6ex]\textstyle\rightarrow \\[-0.6ex]\textstyle\rightarrow}}\cdots\Big).$$
To see this last point, since $R\Gamma(X_\tau,-)$ is a right adjoint, it will suffice to see that
$$\mathcal F=\lim\Big(Rf_*f^*\mathcal F\rightrightarrows Rf_{1*}f_1^*\mathcal F\mathrel{\substack{\textstyle\rightarrow\\[-0.6ex]\textstyle\rightarrow \\[-0.6ex]\textstyle\rightarrow}}\cdots\Big),\tag{$\ast$}$$
in $\mathcal D^{\geq0}(X)$, where $f_0=f:U\to X$ and $f_n:U_n=\underbrace{U\times_X\ldots\times_XU}_{n+1\text{ times}}\to X$. Replacing $\mathcal F$ by a bounded below complex of injectives in $\operatorname{Mod}_\Lambda$, we may drop the $R$, since the maps $f_n^*$ preserve injectives as the $f_n$ belong to our topology, and our task is to compute the homotopy limit of the diagram appearing in $(\ast)$. Such homotopy limits are called totalizations, and much has been written about computing totalizations of diagrams of chain complexes ([nLab][2], https://mathoverflow.net/questions/194010/reference-for-homotopy-colimits-of-cochain-complexes-via-totalization-of-dou, https://mathoverflow.net/questions/361064/reference-for-homotopy-colimit-total-complex/, §15 of [this article by McClure & Smith][3]). Unfortunately none of these accounts are really satisfactory, but the gist of it seems to be that given a cosimplicial map $\Delta\to\operatorname{Ch}$, we form the normalized Moore cochains in $\operatorname{Ch}$; i.e. a double complex $K^{ij}$, and then take the total complex. (I restrict to non-negative chain complexes because I don’t know if this recipe holds in the unbounded case.)

If we think about the complexes appearing in $(\ast)$ in the horizontal direction and the cosimplicial maps in the vertical direction, and we think of $\mathcal F$ as a double complex concentrated in row 0, then we have a map of double complexes $F\to K^{ij}$ that we wish to see induces a quasi-isomorphism after totalizing. Now it’s helpful to know that normalized cochains are quasi-isomorphic to un-normalized ones ([019H][4]), so by the spectral sequence associated to the filtration by columns it will suffice to show that the augmented relative Čech complex
$$\mathcal F\to f_*f^*\mathcal F\xrightarrow{d^0-d^1}f_{1*}f_1^*\mathcal F\xrightarrow{d^0-d^1+d^2}\cdots$$
is acyclic when $\mathcal F\in\operatorname{Mod}_\Lambda$ and nothing is derived. As $f:U\to X$ is a covering map, it will suffice to check this after pulling back to $U$, but now we’re in the situation of forming the relative Čech complex associated to a map which admits a section: the resulting complex is homotopic to zero ([06X6][5]).


  [1]: https://mathoverflow.net/questions/426440/derived-functors-out-of-an-unbounded-derived-infty-category/426879
  [2]: https://ncatlab.org/nlab/show/homotopy+totalization#chain_complexes_of_abelian_groups
  [3]: https://arxiv.org/abs/math/0402117
  [4]: https://stacks.math.columbia.edu/tag/019H
  [5]: https://stacks.math.columbia.edu/tag/06X6