The structures of those two examples are quite different. They do both include inversions, but those inversions differ in both type and meaning.
Hallowed be Thy name.
This is an example of subject-dependent inversion. The dependent (meaning the verb's argument) is the complement "hallowed", and the subject is "Thy name". Without the inversion, the sentence would read "Thy name be hallowed."
The sentence has a meaning similar to "may your name be hallowed" or "let your name be hallowed" or "your name must be hallowed". The form "be" marks a subjunctive mode, and the inversion simply supplies a shift in emphasis or serves to provide a more poetic rhythm.
Be it ever so humble, . . .
There is no subject-dependent inversion here. The beginning of the earlier example is a subject complement. The beginning of this clause is the verb itself. This is an example of subject-operator inversion. Without the inversion, the clause would read "it be ever so humble".
The combination of subject-operator inversion and subjunctive mode mark this example as a dependent clause. The clause has a meaning similar to "Even if it is ever so humble, . . . " or "Although it might be ever so humble, . . . ". It's use and meaning do not much resemble those of "May it be ever so humble." It's a necessarily dependent clause, unable to form a sentence on its own.
Modern dialects don't make much use of this subjunctive subject-operator inversion. Common examples like "Be it ever so humble, there's no place like home" and "Be he live or be he dead, I'll grind his bones into my bread" are linguistic fossils. A similar thing can be said for third-person subjunctive imperatives. They exist in legal jargon and religious expression, but seem to have little currency in everyday speech.