Non-Fermi Liquids

In Fermi liquid theory, the assumption made (to my knowledge) about the status of quasi particles from the field theory point of view is that the self energy $$\Sigma$$ in the interacting theory does two things, namely modify the structure of the propagator, changing the mass of the quanta (here mass renormalization has then the effect of defining the mass of the quasiparticle) and the imaginary part determines the stability of these quasiparticles. In what sense and to what extent to either or both of these assumptions break down for non Fermi liquids?

First, it must be noted that Non-Fermi liquids are anything that are not Fermi liquids. As such, universal statements for all Non-Fermi Liquid behavior are not really possible.

"...the self energy Σ in the interacting theory does two things, namely modify the structure of the propagator..."

I'll assume that you are talking about the retarded propagator, $$G_R$$. Note that this is in general, a function of both energy $$\omega$$ and momentum $$k$$. ie, $$G_R(\omega, k)$$

The relation $$G_R = (G_0^{-1} - \Sigma_R)^{-1}$$, where $$G_o^{-1} = \omega - \epsilon(k) +\mu$$ is the inverse of the free propagator (defined for a Grand canonical ensemble at chemical potential $$\mu$$) and $$\Sigma_R$$ is the (full) retarded self energy, is an exact one. All we have done is identify repeating blocks in the full perturbative series, and collect those into one term ; $$\Sigma$$. The relation is formal, and always holds.

"...the imaginary part determines the stability of these quasiparticles..."

Actually, many phenomena described by the term Non-Fermi liquids don't even have well defined quasi-particles.

From above, $$G_R(\omega,k) = \frac{1}{(\omega +\mu) - (\epsilon(k) + Re\Sigma_R) -iIm\Sigma_R }$$, where it can be motivated that $$-Im\Sigma_R$$ is a positive number (basically, because we are dealing with a retarded object).

The quasi-particle picture is valid when $$-Im\Sigma_R \to 0^{+}$$. This is what happens in the standard Landau Fermi liquid. The consequence of this is that $$G_R$$ has a sharp peak (around the zero of the real part of the denominator). The location of this peak is called the quasiparticle energy ; the system is said to have particle-like excitations, with lifetime $$~ \frac{1}{-Im\Sigma_R} \to \infty$$ (ie, longlived quasiparticles)

Compare this with the case when $$-Im\Sigma_R \to 0^{+}$$ does not hold true. In this case, we end up with a broad peak ; there are no well defined quasiparticles in this case.

This is exactly the case in quantum critical phases ; there are no well defined quasiparticles. However, there is still a fluid. This then falls into the category of Non-Fermi Liquid