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This question is related to another I just posted, but it is actually a different question: if Black Defends the Tartakower QGD without an early ...h6, and I am an Anti-Tartakower–Makogonov–Bondarevsky vartiation player (5... h6 6. Bxf6 Bxf6), should I take an early Bxf6 nonetheless, by my own initiative, or should I keep my Bishop on g5?

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It depends on your own tastes, but I'd recommend not taking. Taking after ...h6 is justified because it essentially costs Black a tempo (i.e., playing ...h6). If you take without provocation then no one's losing tempi, you're just making an exchange.

If you really like trading on f6 then go ahead, but objectively you shouldn't have any advantage.

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Be careful that you don't end up a tempo down in a known line, otherwise it is good to keep all your options open. h6 can benefit Black in giving luft to a castled king but overall it is a slight weakening of his position, so you should be encouraged to wait for Black to ask the Bishop it's intentions.

Overall, unless you have a concrete reason to take the knight you should keep the Bishop.

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I think what's important here is a conceptual understanding of the openings involved. What you're essentially asking is whether or not Bxf6 is worth considering in something like the Orthodox Defense variation of the QGD or some Semi-Slav variation, and what are its differences compared to the the Bxf6 move featured in the Anti-Tartakower variation of the Neo-Orthodox.

Firstly, one of the key reasons for which people may play the Neo-Orthodox rather than the Orthodox (or Semi-Slav) is that the latter ends up being more passive and cramped than the former, while still being solid. However, it features a key weakness: it uses a tempo to move the h-pawn. Even though White generally prefers to maintain the bishop (at least for a while) and to maintain the pressure in the position, the Anti-Tartakower nevertheless becomes coherent in light of this, as it takes advantage of a "wasted" tempo. Additionally, it avoids some of the more theoretical lines of the Tartakower. However, despite this, it's still generally considered a slightly worse try for White compared to the main line with Bh4.

In the Orthodox Defense, however, Black generally plays 5...O-O or 5...Nbd7. If Black plays 5...Nbd7, then 6.Bxf6 is a poor move, as it reduces White's pressure on Black's position, it makes Black's position less cramped, and it improves Black's knight on d7, all of that without even needing to invest a tempo on the h-pawn. If Black plays 5...O-O, then 6.Bxf6 is transposing to the Anti-Tartakower (which, as discussed, is already regarded as arguably not being the best try for White) but where Black has an extra tempo (as h6 wasn't played). Therefore, you can quickly tell that it must be an objectively poor choice for White. And this allows us to also address the differences between Bxf6 in the Orthodox and Bxf6 in the Neo-Orthodox: Black having an extra (and highly valuable) tempo to get his queen-side minor pieces out. Therefore, typical Anti-Tartakower plans such as playing Qb3 and trying to put pressure on the queenside will not be as effective. Therefore, while Bxf6 in the Orthodox is probably not outright losing, it almost certainly sacrifices White's opening advantage.

It's also worth noting that the same applies in the Anti-Meran Gambit of the Semi-Slav (1.d4 d5 2.c4 c6 3.Nf3 Nf6 4.Nc3 e6 5.Bg5). If Black plays the Moscow Variation (5...h6), then 6.Bxf6 is a good continuation for White. If, however, Black opts for something else (e.g. the Cambridge Springs Defence starting with 5...Nbd7), then 6.Bxf6 is no longer considered as good.

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