What interaction breaks $SU(3)$ flavor symmetry? I know that flavor symmetry is only approximate, broken since the u,d, and s quarks have different masses. Also that trying to extend $SU(3)$ symmetry to $SU(4)$, $SU(5)$, or $SU(6)$ symmetry is useless, because the masses of the c, b, and t quarks are very different from the lighter quarks.
I understand that isospin symmetry, which is an $SU(2)$ subgroup of $SU(3)$ flavor symmetry, is broken by the electroweak interaction. That is, if u and d quarks only interacted via the strong interaction, then isospin symmetry would be exact. In some sense, then, the different masses of the u and d quarks are due to the electroweak interaction.
What about $SU(3)$ flavor symmetry? If the u, d, and s quarks interacted only via the strong interaction, would it be exact? Is the electroweak interaction ultimately responsible for the symmetry breaking?
Or are interactions with the Higgs boson somehow involved? This thought occurred to me because the different masses of the three quarks are always cited as the reason for the symmetry breaking.

To check my understanding of Zachos' answer and this stackexchange answer, I will now attempt to express their content in my own words.
The Hamiltonian for quark interaction consists of several terms, among them:

*

*"Characteristic" strong interaction terms

*Higgs Yukawa terms, aka mass terms

*Electromagnetic and weak interaction terms

The characteristic strong terms commute with the flavor operators, so if they were the only terms, flavor symmetry would be exact. The flavor operators commute with neither the mass terms nor the electoweak terms, so both of these break the symmetry.
Unlike SU(3) flavor symmetry, SU(6) flavor symmetry is useless for certain kinds of (approximate) calculations, but it has other uses.
 A: 
That is, if u and d quarks only interacted via the strong interaction, then isospin symmetry would be exact. In some sense, then, the different masses of the u and d quarks are due to the electroweak interaction.

There  is an overburden of misconceptions to unpack. Indeed, also EM breaks the isospin SU(2), in the conventional sense of electric charge not commuting with isospin generators;  indeed, the Higgs Yukawa couplings dictate/encode the different masses of the quarks, which underlie all flavor symmetry breakings, explicitly and irrevocably. (Even though they are packaged together with the  weak interactions, Higgs interactions are not a part of the gauge structure of the weak ones. But this is subtle, and can wait for fuller competence in the SM.) In any case, they do the breaking asked about in the question title.
But much of what you think you are summarizing is flat wrong.
The masses of the u and d are not due to EM. If one switched off the EM interactions, isospin symmetry would still not be exact.
The strong interactions do not ultimately respect flavor symmetries such as isospin SU(2), SU(3), just as they don't respect SU(4), SU(5), SU(6) (which, however, are not useless at all : they lead to hadron classification schemes which are quite valid and useful.)

*

*A bit of care here, given your comment: the strong interactions do treat all quarks the same, and the color charges commute with the (non-conserved) flavor charges. But the strong hamiltonian, involving only strong interactions and no EM or EW interactions, is not flavor symmetric: the quark masses are already different, so the resulting hadrons are not flavor degenerate!

The point of isospin is that the explicit breaking parameter of it  is small,
$$
(m_d-m_u)/\Lambda_{QCD}\sim 1\%,
$$
and the point of SU(3) that it's breaking parameter is also small,
$$
(m_s-m_d)/\Lambda_{QCD} < 1/2 .
$$
The first is an excellent approximate symmetry of the strong hamiltonian, and the second a passable/useful approximate symmetry.
Since the masses of these light hadrons are set by the chiral symmetry breaking of QCD, the above Λ, you may tentatively
"defocus" from this small explicit breaking in studying hadronic couplings, and only consider it in systematizing hadron multiplet mass splittings, all in the context of hadronization, driven by the strong interactions.
