It doesn't have the right weak equivalences! An equivalence of $(\infty,1)$-categories is not just a functor inducing an equivalence on homotopy categories. For example, the functor $C \to ihC$ is not an equivalence if $C$ is not in the image of $i$.

The reference in Higher Topos Theory for the positive results is Section A.3.7.

See section 2.3 of math.uni-bonn.de/~mgroth/groth_monoidal.pdf.

Flatness is equivalent to vanishing of all higher Tor groups, which are the same as the homology groups of derived tensor products, and derived tensor products are homotopy pushouts of commutative dg-algebras. A homotopical characterization of étaleness or smoothness can be given using the cotangent complex. See any of the usual references on derived algebraic geometry, e.g. Lurie's thesis or HAG II by Toen-Vezzosi. This all goes back to Quillen, whose original notes on homotopical algebra and the cotangent complex are also very good.

Somoth/étale morphisms can be lifted (Zariski-locally on the source) along closed immersions (EGA IV_4, 18.1.1).

@domenicofiorenza They must be inverse to each other because they form an adjunction. This is a general category-theoretic fact, of course.

@domenicofiorenza $\Sigma$ is an equivalence by the definition of triangulated category...

Outside characteristic zero, you have to use $E_\infty$-dg-algebras instead of strictly commutative dg-algebras. See Tyler Lawson's answer here: mathoverflow.net/a/23885/2503. (Oops, Denis beat me to it.)