If Levirate marriage conflicts with the law of the land, eg laws of prohibited affinities for marriage, do (Orthodox) rabbis still solemnise Levirate marriages? If not, when did they stop doing so, and under whose decision?
To my knowledge, Levirate marriage, or “Yibum”, never came into conflict with modern laws, simply because the alternate practice of “chalitzah”, which frees the man and woman without marriage, took precedence.
The Mishnah in Bechoros 13a reads: the Mitzvah of Yibum takes precedence over the Mitzvah of Chalitzah — at first, when they were thinking properly about the Mitzvah. Now, when they are not thinking properly about the Mitzvah, the Mitzvah of Chalitzah takes precedence over the Mitzvah of Yibum.
Abba Shaul says in Yevamos 39b (an entire tractate devoted to Levirite marriage), “someone who brings in his Yevama (the woman previously married to his deceased brother) for her beauty, to have a wife, or for any other purpose, is as if he violated an immoral relationship, and in my eyes the child is close to being a mamzer (a product of a forbidden relationship, e.g. incest).”
While others disputed this opinion, the universal custom has long been to perform chalitzah instead of Yibum. You can read more about this here. I hope this has answered your question!