Let me formulate as an answer what I know about the question. I'd be happy to be corrected if something said here is inaccurate or incomplete. I assume throughout dimension $d=4$.
First of all, it seems to me that there is no definition of what $\Phi^4$ theory should mean in rigorous terms. Thus it is presently impossible to disproof the existence of $\Phi^4$ theory.
On the other hand, I think that there is consensus about how to judge a claimed existence proof of $\Phi^4$ theory: it would mean to have a Hilbert space and a family of operators rigorously defined on it, such that one can derive from it in a formal way (not necessarily rigorous, but valid on the level of the standards of the common textbooks on QFT) an asymptotic expansion of the S-matrix agreeing order by order with the textbook expansion.
The accepted mainstream view is that the existence of a Landau pole at very high energies in the approximate (low order perturbative or low resolution lattice) treatments of $\Phi^4$ theory, together with the most likely triviality of the continuum limit of the lattice version imply (in some unspecified roundabout way) the nonexistence of $\Phi^4$ theory. Useful papers representing this view are
Discussions of the existence problem (dubbed ''triviality of $\Phi^4$ theory) that take a less decisive stand are in
On the other hand, there is some evidence for existence of $\Phi^4$ theory or QED. None of these is accepted mainstream physics, but I haven't seen convincing reasons for dismissing these signs.
1. Klauder (well-known for work on coherent states in quantum optics) proposes alternative renormalization schemes that seem to work to some extent:
2. The so-called PT-symmetric QM formalism
gives a construction of a Hilbert space for the Lee model with a ghost mode, which exhibits a Landau pole in perturbation theory. There is also a perturbative QED version:
3. There are non-perturbative constructions for approximations of QED, where the Landau pole disappears, e.g.,
Note that 4D Yang-Mills nonabelian gauge theory, which is asymptotically free, also has a Landau pole in perturbation theory, and - worse than for QED and $\Phi_4^4$ - it is located at experimentally relevant energies! See https://arxiv.org/abs/1311.6116. But this is not taken as an indication of nonexistence of 4D Yang-Mills theory; instead (see the survey by Shifman, https://arxiv.org/abs/hep-ph/9802214) it is only taken to mean that at low energies, a nonperturbative treatment is called for.