Was there ever an axiom rendered a theorem? In the history of mathematics, are there notable examples of theorems which have been first considered axioms?
Alternatively, was there any statement first considered an axiom that later has been shown to be derived from other axiom(s), therefore rendering the statement a theorem?

The most famous example I know is that of Hilbert's axiom II.4 for the linear ordering of points on a line, for Euclidean geometry, proven to be superfluous by E.H. Moore.  See this wikipedia article, especially "Hilbert's discarded axiom". https://en.wikipedia.org/wiki/Hilbert%27s_axioms
In the article of Moore linked there, it is stated that also axiom I.4 is superfluous.
http://www.ams.org/journals/tran/1902-003-01/S0002-9947-1902-1500592-8/S0002-9947-1902-1500592-8.pdf
