Horace Jeffery Hodges over at Gypsy Scholar was exploring Hebrews 9:23. The key question I believe he is asking is what it means to cleanse the heavenly sanctuary: "Therefore, [it was] necessary for the examples of the things in the heavens to be cleansed with these [things], but the heavenly [things] themselves with better sacrifices than these." I personally believe that heavenly [things] here refers to the heavenly holies, the heavenly Most Holy Place.
So the age old question is this--why would the heavenly sanctuary need cleansed? It's in heaven. Lincoln Hurst has given us a clarification that alleviates the problem a little, but not completely--this is about inauguration of a sanctuary. But the reason inaugural sanctuaries need cleansed is still to make them pure and holy. Harold Attridge and others have suggested perhaps what Hebrews is pretty much talking about is the cleansing of the conscience.
Here is my personal sense of what's going on here in Hebrews. With language that is serving rhetorical purposes, you have to get a full picture of what is going on to really understand the significance of the language. So for Hebrews, the basic rhetorical point is that Christ's death has removed any necessity for the Levitical system and its sanctuary. We can debate whether the sanctuary is already gone and the author is, in a way, consoling the audience (my position) or whether he is dissuading the audience from using a standing structure.
The author's rhetorical strategy to make this point is to construct a complex metaphor in which every key element of the Levitical system is surpassed by Christ's death. So if the Levitical system had priests, Christ is a priest to end all priests, after the order of Melchizedek (Heb. 7). If the Levitical system had sacrifices, Christ is the sacrifice to end all sacrifices (e.g., 10:14), an "eternal spirit" sacrifice even (9:14).
With regard to the sanctuary, the author drew on existing metaphors that considered heaven the truest sanctuary of God, with the earthly sanctuary modeled after the universe (Philo, Josephus). I personally find no compelling evidence in the argument of Hebrews 8-10 to think that the author sees an actual structure in heaven with two rooms. Rather, I argue that the author sees the highest heaven as a kind of Most Holy Place where God dwells.
So the idea of "inaugurating the heavenly sanctuary" with a better sacrifice does not correspond neatly to one thing because it is part of an overall, complex metaphor. What is being cleansed here? An abstraction. "Inauguration of the heavenly sanctuary" means the commencement of that age of reliance on Christ's death as means of atonement or, in Hebrews' terms, the age of real atonement versus proleptic rain checks looking forward to atonement.
Perhaps I should not even say that the cleansing of the conscience is the closest "literal" meaning of the image, then. The author is not saying that the "literal" heaven needed cleansed. He is simply playing out a metaphor, and when you do that, the image eventually breaks down into unintended resonances.