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 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

http://alumnus.caltech.edu/~callaway/trivpurs.pdf

http://journals.aps.org/prl/pdf/10.1103/PhysRevLett.93.110405

http://arxiv.org/pdf/hep-lat/0009029.pdf

Discussions of the existence problem (dubbed ''triviality of $\Phi^4$ theory) that take a less decisive stand are in

http://sargyrop.web.cern.ch/sargyrop/SDEsummary.pdf

http://arxiv.org/pdf/0806.2196.pdf

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:

http://arxiv.org/pdf/1308.4658.pdf

2. The so-called PT-symmetric QM formalism

http://journals.aps.org/prd/pdf/10.1103/PhysRevD.71.025014

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:

http://arxiv.org/pdf/hep-th/0501180.pdf

3. There are non-perturbative constructions for approximations of QED, where the Landau pole disappears, e.g.,

http://arxiv.org/pdf/hep-th/0505021.pdf

http://arxiv.org/pdf/hep-th/0111152.pdf

http://link.springer.com/article/10.1007/JHEP05(2011)119#