Is the effective Hamiltonian obtained via Feshbach-Fano partitioning closed?

For the Feshbach-Fano setting, you have a Hamiltonian $H$ for the full system. The Schrödinger equation with this Hamiltonian tells you how the full wavefunction $|psirangle$ evolves.

You then partition your system, as hinted at in the OP, into a system and an environment. This is done by introducing projection operators $P$ and $Q$. $P|psirangle$, for example, is then the wave function in the $P$-subspace. There are then also different projected Hamiltonians. For example, $H_{PP} = PHP$ is the Hamiltonian projected onto the $P$-subspace.

Now let us ask the following question: Which Hamiltonian tells us how the subspace wave function $P|psirangle$ evolves? So which Hamiltonian do we have to plug into the Schrödinger equation to get the time dependence of $P|psirangle$?

Our first guess could be $H_{PP}$. However, this cannot be correct since the $P$-space wave function also interacts with the $Q$-space environment. The answer is instead the effective Hamiltonian, which $H_{PP} + textrm{some term accounting for the interaction with $Q$}$ (note that compared to the linked wiki article, I have $P$ and $Q$ swapped).

Let me answer the questions in the OP more directly.

"Can a resulting effective Hamiltonian be considered to describe the closed quantum system?"

No, it rather describes the evolution of the open sub-system by including the interaction with the environment.

My intuitive answer is yes because you are removing the "environmental" effect?

You are thus not "removing" the environmental effect, the point of the effective Hamiltonian is rather to include the enviroment interaction without having to keep track of the environment explicitly. If you wanted to remove the environment, the Hamiltonian to use would be the projected sub-space Hamiltonian $H_{PP}$ on its own.

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Let me also point you to some further reading by mentioning key words. The Feshbach-Fano partitioning method is particularly popular in atomic theory (such as quantum chemistry) and in scattering theory. In these fields one is often still interested in what the environment does in one form or another and the partitioning is introduced to simplify the problem or introduce approximations (note that this is a very rough take on this big subject and probably a gross oversimplification).

If you are interested really kicking out the environment to see what a physical subsystem does, you may find the formalism of open quantum systems more relevant. There, you "trace out" the environment to find the evolution of your subsystem. This is closely related to the Feshbach-Fano partitioning technique, but to have a physical notion of the subsystem, one works with density matrices instead of projected wave functions.