Definite or indefinite article in "the/a devil's advocate"

He is the Devil's advocate.

This is the classical expression. The term was used by the Catholic Church (from 1587 until the office was abolished, in 1983) for the canon lawyer who was supposed to argue against proposals for canonisation, i.e. adding someone to the official list of saints, the canon. The purpose of these arguments against canonisation was to test the strength of the arguments for canonisation as brought forward by God's Advocate.

Syntactically, it is ambiguous whether the modifies Devil or advocate; however, in this case it must modify Devil. That is because the Devil normally requires the definite article if you are referring to the one and only Christian Devil, which is the case here. The definite article can sometimes be left out, but that would be ellipsis; in that case, however, advocate shouldn't have an article either, because Devil requires it while advocate doesn't. Compare the following sentence:

He is Cleopatra's advocate.

He is the Queen's advocate.

Being someone's x usually doesn't require an article before x. Whose advocate is he? The Devil's! It would be odd to add the article where it is normally left out (with advocate) while omitting it where it is normally used (in the Devil).

He is a devil's advocate.

The indefinite article sounds less idiomatic. The article the as above could be left out in casual use; but then it would sound odd to use a phrase almost identical to the full classical expression the Devil's advocate, having merely swapped one article for the other. If you mean to say that a specific person answers to this description, use the; if you were mentioning the general concept of being a devil's advocate, you could very well use a.

He is playing devil's advocate.

Here the article is dropped in a casual manner, and the phrase is used loosely in a slightly changed environment: this is how the phrase is most often used.

All three uses are metaphorical, meaning, as Cerberus points out, someone arguing for the obvious negative/unexpected side. It is never meant literally (even historically), so the article or lack thereof is about the entire consitituent 'devil's advocate', 'the' for a particular person arguing in that manner, 'a' for an unspecified one, and no article for filling the general concept.