One should not rent a bath-house to a non-Jew because this will look as if they are performing work on behalf of the Jew on Shabbat.
By implication, renting a bath-house to a Samaritan should be permitted. But while Samaritans observe the Sabbath and do not work on it, they do work on intermediate days of the festival. The assumption here is that Jews do not work on these days and thus people might think that he is working on behalf of the Jew.
The answer is that Jews also are allowed to work in a bath-house on hol hamoed by heating up the water.
A Jew is allowed to rent, at least outside of Israel, a field to a non-Jew, even though non-Jews will certainly work there on Shabbat. Why are we not concerned lest people think that non-Jews are working for Jews on Shabbat? The answer is that people will think that he is a tenant—someone working for a share of the crops. It is permitted to have such an arrangement.
But then why not say the same thing about the bath-house—people will think that the non-Jew has a tenant relationship with the Jew, and works the bath-house for a share of the profits? The answer is that such business relationships are not typical for bath-houses.
Today’s sugya contains another baraita in which R. Shimon b. Elazar discusses renting to a Samaritan.
A Jew should not work his fields on the intermediate days of the festival (hol hamoed). But Samaritans do not seem to observe this rule. So if one rents a field to a Samaritan and others think he is working the field on behalf of the Jew, they will say that the Jew is working his own field on hol hamoed.
If it is forbidden to rent a field to a Samaritan, it would seem that he would say that it is permitted to rent a field to a non-Jew. The reason is that people will assume that this a tenancy arrangement. But if this is so, why not assume the same thing about the Samaritan. Why assume that people will think he is working the field on the Jew’s behalf?