Grammatically, the valediction of a letter plus the underlying signature line form a single sentence. I have been trying to go through all possibilities that seem valid in German, but they all boil down to the valediction being an expression wherein the signing entity is the subject; the subject of a sentence typically requires nominative case and thus nominative case is used in the signature line. Compare:
Alles Gute (wünscht dir)
Dein Freund Sepp.
Mit der Hoffnung auf rege Teilnahme an der Hauptversammlung grüßt
Der Vorstand.
The second example is a bit made-up; more often there will be a first-person subject, to which the final line can be considered an apposition:
Ich wünsche Ihnen noch einen schönen Urlaub
Der Personalverantwortliche
In all these examples I used masculine nouns because they can be assigned one of the four cases unambiguously, which is not the case for feminine or neuter nouns.
The second frequent example you mentioned in your question are objects (e.g. statues) which are to be seen as a gift towards somebody; like the phrase on the Reichstag. This can be seen as partial sentences formed around the verb widmen:
(Der Kaiser widmet dieses Gebäude) Dem deutschen Volke
(Diese Statue{acc} widmet) Dem Landesvater […] sein treues Volk{nom}.
I’m not aware of other cases but that is because I don’t usually think about them. In any case, asking for all of them would be too broad; you should ask about them if you encounter them.
