In general, having constant mean curvature is not preserved. This can be seen by a direct computation, or by looking at examples: The round metric on the 3-sphere without a point is conformal to flat 3-space. The Clifford torus in the 3-sphere (even without a point) is minimal, hence of constant mean curvature, and embedded. After stereographic projection the Clifford torus is not of constant mean curvature, as you can prove by computation, or from Alexandrovs theorem: There are no compact embedded CMC surfaces in euclidean 3-space besides the round sphere.

The functional which is invariant under conformal change of the ambient metric is the Willmore functional (and not the area functional):
$$\mathcal W(\phi)=\int_\Sigma (H^2-K+\bar K )dA,$$
where $H$ is the mean curvature, $K$ is the curvature of the induced metric, $\bar K$ is the sectional curvature of the tangent plane of $\phi$ and $dA$ is the area form, all w.r.t. to the metric $g.$ But in fact, the integrand $(H^2-K+\bar K )dA$ is invariant under conformal changes of the metric.
This was known by Sophie Germain, Blaschke and others for Moebius transformations, but first shown in the general case by "B.Y. Chen, Some conforaml invariants of submanifolds and their applications, Bol. Un. Mat. Ital, (1974)", see also Joel Weiner: On a probelm of Chen, Willmore et.al., Indiana Math Journal (78).

As a first consequence, being a Willmore surface, i.e., a critical point for the Willmore functional, is invariant under conformal changes of the metric. It is possible to define the Willmore functional for surfaces into the conformal 3-sphere without any reference to a Riemannian metric on the 3-sphere, see e.g. the work of Burstall, Pinkall and Pedit.