The contradiction between Gell-mann Low theorem and the identity of Møller operator $H\Omega_{+}=\Omega_{+}H_0$

The Gell-Mann Low theorem applies only to eigenvectors, i.e. to the discrete part of the spectrum. Hence it does not apply to scattering states. The latter are not eigenvectors since they are not normalizable. Your formula for $\Delta E$ is meaningless for them since the inner product on the right hand side is generally undefined unless $\psi_0$ is normalizable.

[The equation for the Moeller operator] ''says the energy of scattering state will not change when you turn on the interaction adiabatically.'' No. It only says that $H$ and $H_+$ must have the same total spectrum; it says nothing about energies of individual scattering states.

Moreover, a more rigorous treatment (e.g. in the math physics treatise by Thirring) shows that your equation holds at best on the subspace orthogonal to the discrete spectrum (which almost always exhibits energy shifts), and that certain assumptions (relative compact perturbations) must be satisfied that it holds on this projection. These assumptions are not satisfied when the continuous spectrum of $H$ and $H_0$ is not identical, e.g., when $H_0$ is for a free particle and $H$ for a harmonic oscillator or a Morse oscillator, or vice versa.

The second and third statement seemingly are not necessarily true without any further assumptions: If one takes the trivial example $H_0 = H_i$, then the eigenstates don't change, there are neither more nor less eigenstates, and even the continous energy spectrum is changed: All energies are multiplied by $1+e^{-\epsilon |t|}$.