Do authors always have final approval over what gets published?
In the answer you are referring to there is nothing to say that approval is not required. The issue is rather that an editor can request certain things can be removed from the paper so the author cannot demand everything written is finally printed. Editors have the right and obligation to weed out materials that are, for example, offensive just as we can flag posts that contain such phrasings on Stackexchange. Jokes, which was the point of the post you referred to, are not edited out by any means as a rule but can be misunderstood or considered offensive and might be more likely to be edited out than other material during the review and editorial process. In such a case, the author will certainly be part of the process and asked to revise or accept changes. the point here is that the editor has the final say through approval of the paper as well as a responsibility to make sure the publication is free from material that violates, for example, ethical of the publisher. As an example, in Sweden, where I reside, I as an editor am responsible by law for what is printed in "my" journal. This responsibility may lie with the publisher in other countries.

So, authors will definitely be part of the revision process and sign off on final proofs, it is just that some changes may not be subject to additional discussion and will have to be accepted. Does this happen often? Not at all. In my case, I have had to deal with this in heated "Letters to the Editor" where authors have been in a written, published, exchange over issues in a published paper.

Edit (to expand based on earthling's comment): Basically all changes except typos and changes to adhere to journal or publisher style are subject to an "OK" from the author (although it would not hurt to also get an OK on these). The publishing of a paper is a joint venture between an author and the journal, so with a functioning working relationship both parties should agree on changes. But, again, some changes are non-negotiable, which does not mean they can be imposed without the knowledge of the author who always has the option to go elsewhere to publish. 

So changes can come in three forms, (1) those that can be changed without consent (e.g. typos), (2) those that must be made without discussion (subject to ethical considerations) and (3) those that can go either way after, say, clarification (most of the normal changes). In  cases 2 and 3 the author need to approve the changes it is just that for case 2 the option might be accept or withdraw the paper.