What would happen if someone lacked the genes to produce RNA polymerase II?

Then the person won't be able to transcribe DNA into mRNA, therefore there will be no RNA to be translated to protein and the person won't have proteins and die! The End.

The fertilized egg representing the first moment of the person's existence would still be able to synthesize rRNA, tRNA, siRNA etc, so would possess all the cellular machinery for protein synthesis, but the inability to transcribe mRNA would render it incapable of protein synthesis, so the egg would never develop into a person. The only way that cell could even survive, let alone reproduce would be a continuous and tightly orchestrated regimen of mRNA transfusions, which is beyond current technology although, in principle, gene therapy to replace the missing RNA Pol II gene is possible.