I think the house very clearly signifies death.
We first see the colonial-style house in Season 4's "Calling All Cars," when Tony has a disturbing dream of approaching it, first with Ralphie and then alone, and then apprehensively calling out to a vague, shadowy woman figure on the stairs. This figure strongly resembles Livia.
The house as seen in the dream:
The next appearance of the house is in Season 5's "All Due Respect," in the scene where Tony murders his cousin Tony B. It is the house Tony B. was staying at, and just about to enter when Tony blasted him with that shotgun. We see it well in a wide shot:
Significantly, we next see the house at the end of Tony's coma dream/trip, in Season 6's "Mayham":
Remember that the deceased Tony B is the one urging Tony to enter the Inn, and there is an ambiguous Livia-like female figure standing inside by the doorway. This figure seems almost identical to the Livia figure in Tony's "Calling All Cars" dream.
The last sighting of the house is in the final episode:
The choice of that house to be behind Tony is no coincidence. I believe it represents death, as suggested in "Mayham," and along with the orange cat it looms behind Tony as a death omen in his final moments on the show.
This is only a brief analysis, possibly to be added to later. I just haven't seen anyone connect all these almost identical-looking and death-infused houses before, and I think the connection is important.