Theorem. Obviously, one should never use "obviously".
Proof. Either the "obvious" thing is not obvious after all (in which case, saying/writing "obviously" is wrong) or the "obvious" thing is indeed obvious, and saying/writing "obviously" does not add anything to the matter. In both cases, one is better off not saying/writing "obviously". Q.E.D.
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@jorgeapenas Obviously you’re right, but people may use “obviously” to earnestly signal to the listener that while stating some info is important to their point, they are not assuming the listener doesn’t know it. And obbbbviously the word could never be used sarcastically to, like, signal that something is a commonly-held but wrong belief.
@psmaldino Obviously, that is the reason why we keep using "obviously".