I am unclear how the inverse mean curvature flow starts with a minimal surface. If there is some point $p$ with mean curvature $H(P)=0$, how should we treat this problem and let the flow start?

It isn't clear exactly what sort of initial conditions you're requiring. The difficulty with minimal initial conditions is part of the reason why it was an amazing result when HuiskenIlmanen constructed a "weak inverse mean curvature flow" which (1) Can start at a minimal surface (technically, for certain things to work nicely, it should be outerminimizing) (2) Exists for all time in an asymptotically flat manifold. AND (3) Still satisfies Geroch monotonicity, i.e. the Hawking mass is monotone along the flow. That one could find a "flow" which satisfies (1) and (2) while still keeping (3) is incredible. Their paper is very readable, although quite long, so I'd recommend that you take a look at it, rather than I try to explain the ideas here. EDIT: I've added some more information below. Its not totally clear what your motivation for the question is; if you add more information perhaps I can answer your question better. It is sometimes possible to define a classical flow which starts at a minimal surface. For example, in the Riemannian Schwarzschild metric $$ g = \left(1+\frac{2m}{r}\right)^4 \delta, $$ on $\mathbb{R}^3\setminus \{0\}$, there is a inverse mean curvauture flow $\Sigma_t = \{r(t)\}\times \mathbb{S}^2$ defined for $t>0$, where $\lim_{t\searrow 0} r(t) = m/2$. I'll leave it to you to compute the associated ODE (HINT: the easiest way is to use the fact that $\Sigma_t=e^t\Sigma_0$. Be very careful with what I mean here for $t=0$. In particular, the PDE is not satisfied for $t=0$, just $t>0$. Here are some results about the classical flow: http://www.ams.org/mathscinetgetitem?mr=1064876 

