I guess that when there is surplus of methionine in the cell it is incorporated in the TCA cycle as a succinyl CoA, with cysteine as a by-product. But now the cell has the surplus of cysteine. What does it do with it? How does it remove excess sulphur from the cell? Would that process somehow influence NAD+/NADH balance?
In humans, cysteine first fills the glutathione pool (which itself is a reservoir for cysteine, apart from reduction equivalents and biotransformation). Excess cysteine can go three ways,
Since cystathionine gamma lyase which (its L-cysteine desulfhydrase activity) catalyses 2 and 3 has homologues in yeast, my guess is that H2S is produced there too. Further oxidation of H2S happens through an ancient mitochondrial pathway which is highly conserved as well, yielding sulfite and sulfate. Sulfate then makes for a bit acidic environment but else is non-toxic.
I have linked to reactome.org entries where you can find references. Filling up the yeast's glutathione pool (if it happens) would move the NAD+/NADH balance to the NADH side, i.e., fill up reduction equivalents, too.