The sentence Has he any shame? is not a productive construction in my American English. If, however, the example is changed to Have you no shame?, then it works perfectly well for me. But in such a case, the sentence is likely frozen, that is, fully lexicalized. That subject-verb inversion is generally not productive with lexical have can be seen by constructing random examples (as suggested by user6726 in the comments):
(1) *Has Frank a car? vs. Does Frank have a car?
(2) *Have they any pets? vs. Do they have any pets?
(3) *Had he an excuse for not helping? vs. Did he have an excuse for not helping?
One can probe for the reasons why certain limited cases of subject-verb inversion with lexical have are, although archaic, still encountered in English. My guess is that the construction hearkens back to a earlier stage of English when the V2 principle (verb second) of Germanic was more robustly present. Note in this regard that modern German forms all such questions with subject-verb inversion of the finite verb, regardless of whether the finite verb is an auxiliary or a lexical verb, e.g. Hast du etwas zu trinken?, lit. 'Have you something to drink?'.
The reason inversion has survived somewhat with lexical have, but not more generally with all standard content verbs, is precisely due to the complete overlap in form of auxiliary have and lexical have. Such overlap exists only with one other verb, namely with do, that is, there is auxiliary do and lexical do. Hence a related question is why there seems to be an absence of archaic cases of subject-verb inversion with lexical do, e.g. * Does he always the work?.