I'm not sure to what degree, but it seems that the inscription on the last door bears some resemblance to the Gate of Hell in Dante's Inferno. In "House of Four Doors (Part 1)," there are the lines "Enter in, all ye who seek to find within / As the plaque said on the last door." It seems to be the opposite of "All hope abandon, ye who enter in!" (Canto III, line 9).
The other thing I noticed is the melody to which the middle section ("House of Four Doors / I could live there forever...") is sung. When I first listened to this album two years ago, I wrote about a piano part later in this song that I thought took a phrase from George Gershwin's Rhapsody in Blue. This "House of Four Doors / I could live there forever..." section uses the same melody, and it has more of a resemblance to Gershwin's. The piano part has just the three note diatonic ascent and octave fall, but the vocal melody - like the phrase in the Rhapsody in Blue - has a diatonic ascent after that octave fall. While they're in different keys and the note values are different, the intervals are the same. I still haven't been able to find the Rhapsody in Blue score (I've found a lot pieces that quote it but have been unable to look at the original notation to see how they all compare), but here's some notation I made for this particular section of "House of Four Doors (Part One)":
