The title question is true in the setting of ordinary limits and ordinary schemes; that is, given an inverse limit of schemes along affine maps, the limit still lives in the category of schemes.
I'd like to see that the same thing is true for homotopy limits of derived schemes along affine maps, but I'm having a bit of difficulty making sure that the story remains the same. The ordinary proof is here: https://stacks.math.columbia.edu/tag/01YV, and it seems to me like roughly, every place that a "limit" is mentioned you can replace it with a "homotopy limit," and similarly every "scheme" can be replaced with a "derived scheme."
The argument in the ordinary setting essentially goes that the transition maps $f_{ii'}:S_{i}\to S_{i'}$ are affine if and only if every scheme $S_{i}$ is the relative spectrum $\underline{\mathrm{Spec}}_{S_{0}}((f_{i0})_{*}\mathcal{O}_{S_{i}})$. Then we take the colimit of the pushforwards, and let the candidate limit be the relative spectrum $\underline{\mathrm{Spec}}_{S_{0}}(\mathrm{colim}_{i}(f_{i0})_{*}\mathcal{O}_{S_{i}})$. Check that it is indeed the limit as it satisfies the universal property.
I believe the same proof should work in the derived setting, but I'm a little less comfortable about the moving parts. Does the following work: a map of derived schemes $f_{ii'}:\mathbf{S}_{i}\to\mathbf{S}_{i'}$ is affine if and only if $\mathbf{S}_{i}\simeq\underline{\mathrm{Spec}}_{\mathbf{S}_{0}}((f_{i0})_{*}\mathcal{O}_{\mathbf{S}_{i}})$. Define $\mathbf{S}:=\underline{\mathrm{Spec}}_{\mathbf{S}_{0}}(\mathrm{colim}_{i}(f_{i0})_{*}\mathcal{O}_{\mathbf{S}_{i}})$ and then it is the case that $\mathrm{holim}_{i}\mathbf{S}_{i}\simeq\mathbf{S}$.
Is that true? And/or is this well-known/folklore to the cognoscenti? If not, what is the thing I have to look out for?
