Avodah Zarah, chapter 1, mishnah 9 - the last in the chapter. This mishnah returns to the previous mishnah, which established that there are places and circumstances in which one may rent a house to an idolater, and it comes to qualify and limit that permission.
The wording of the mishnah: "Af makom she'amru l'haskir, lo l'veis dirah amru, mipnei shehu machnis l'socho avodah zarah, shene'emar: v'lo savi to'evah el beisecha" - even in a place where they said one may rent, they did not say to rent it as a dwelling, because he brings idolatry into it, as it says: "And you shall not bring an abomination into your house."
That is: even in a place where renting to an idolater was permitted, renting it out as living quarters was not permitted. The reason is that the idolatrous tenant will bring his idol into the house that the Jew owns and has rented to him, and about this it says "v'lo savi to'evah el beisecha" - a person may not bring an abomination into his house.
The source of the prohibition - a dispute among the Rishonim:
Tosafos and the Rosh: this is a Torah prohibition, that a person may not allow idolatry to be brought into his house, even if he does not live in it himself but rents it out to an idolater.
The Ramban and those who follow him: this is no more than an asmachta. The plain meaning of the verse deals with the prohibition of deriving benefit from idolatry, and the prohibition against an idolater bringing his idol into a house owned by a Jew and rented to him is the Ramban's own concern.
In any event, according to all opinions the matter is forbidden, and a person may not rent out his house to an idolater for living purposes.
Selling as opposed to renting:
Although it sounds counterintuitive, in a place where selling is permitted, one may sell the house to an idolater even for living purposes. The reason: as long as the house remains in the Jew's ownership and he is only renting it out, he may not allow idolatry inside it; but once the house has been sold it is no longer his house, and he bears no responsibility for what the new owner does inside it.
Renting out a bathhouse:
The mishnah closes with a point that sounds connected to what came before, but is in fact only loosely related: "U'v'chol makom lo yaskir lo es hamerchatz, mipnei shehi nikreis al shemo" - and everywhere, one may not rent out to him the bathhouse, because it is called by his name. A person may not rent out the public bathhouse he owns - something like a spa - as a business to an idolater, and in fact to any non-Jew at all. The reason, plainly understood: the rented bathhouse is already known and assumed to be his, and it is called by his name.
The foundation of the matter: it is strictly forbidden for a Jew to own a business and hire people to run it for him on Shabbos. True, there are ways in which a Jew enters into a partnership with a non-Jew, and the business may operate on Shabbos when the non-Jew is the one who receives the Shabbos profits and the like. But the mishnah's assumption is that in its day it was rare and unlikely that the owner of a business, such as a bathhouse, would make the non-Jew a genuine partner who profits on Shabbos; the standard practice was to hire non-Jews merely as workers.
It emerges that even if a person set up an arrangement that is permitted on a technical halachic level, everyone will assume: "This is so-and-so's public bathhouse, and it operates on Shabbos - they heat the water there on Shabbos." For in practice these are only workers, they are not partners and they do not receive the Shabbos profits. Therefore everyone will assume that the owner of the bathhouse is desecrating Shabbos, and that constitutes a desecration of Hashem's name.
This is really no different from a franchise of a coffee shop chain today: the employees who work there on Shabbos afternoon do not keep the profits for themselves; they are salaried workers. Who ever heard of a branch where the employees receive the profits? That is simply not the way the world works. So too the Mishnah takes it as obvious and self-evident that the standard way a bathhouse was run in that time and place was that those working on Shabbos were nothing more than hired employees. Therefore the entire arrangement is forbidden, and not only is there Shabbos desecration involved, there is also a desecration of Hashem's name, and for that reason it is prohibited everywhere.
The practical halachah:
Regarding the bathhouse: Everything depends on what is accepted practice in that time and place. If it were accepted that people enter into partnership with a non-Jew, and that in some fashion the non-Jew runs the business and receives all of the Shabbos profits, it would be permitted, since that is the normal way businesses are conducted. Everything, then, depends on the reality on the ground.
Regarding renting out an apartment: Here matters are more complicated. Certainly outside of Eretz Yisrael, many people own apartments and rent them out without being particular about the religious beliefs of their tenants. There are lenient opinions on the matter, as the Rema cites. Is the reason that people nowadays are not considered idolaters in that serious sense, which is itself a matter of doubt; or perhaps there is at most only a rabbinic prohibition here, according to the opinions we saw above. This subject goes beyond the scope of this lesson.
In any case, this is a practical question: someone who owns an apartment for rent and knows that the prospective tenant is not free of this concern should turn to a rav who is a posek in order to clarify how to proceed.
In summary: In this Mishnah we learned that even in a place where renting to an idolater was permitted, renting for the purpose of a dwelling and residence was not permitted, because he brings idolatry into it, and the Rishonim disagree as to whether this prohibition is from the Torah or merely a supporting allusion. We further learned that selling, in a place where it is permitted, is treated more leniently than renting, since the house leaves the seller's domain entirely. In its second half the Mishnah dealt with the prohibition of renting out a bathhouse everywhere, because it is called by the Jew's name, and its operation on Shabbos involves both Shabbos desecration and a desecration of Hashem's name.