Gittin, Chapter 5, Mishnah 9. This Mishnah discusses the parameters of assisting someone who is suspected of not properly observing halachah - both in the laws of Shemittah and in the laws of tumah and taharah. The Mishnah teaches in which ways indirect assistance is permitted, what direct assistance is forbidden, and which leniencies were established for the sake of peaceful relations.
Lending to a Neighbor Suspected Regarding Shemittah:
A woman may lend utensils to her friend who is suspected of not properly observing the laws of Shemittah - perhaps she keeps Shemittah produce in her home in a stored-away manner, which is forbidden to her, or perhaps she uses the produce after the time of biur, the time when it must be destroyed.
"Nafah u'chevarah v'reichayim v'tanur" - two types of utensils used for sifting and straining flour and the like, as well as a mill and an oven. She may lend them to her, even though the other woman is liable to use them for things that run counter to the laws of Shemittah.
"Aval lo tavor v'lo titchan imah" - she may not sit with her and help her with her own hands to separate the chaff from the good grain, nor may she grind with her. Any action that constitutes direct help is forbidden, since this is direct assistance to those who transgress.
The Wife of a Chaver and the Wife of an Am Ha'aretz:
From here the Mishnah moves to the laws of tumah and taharah. A 'chaver' is a Torah scholar who is careful with the laws of tumah and taharah, whereas an 'am ha'aretz' is not careful with them. The wife of a chaver may lend the wife of an am ha'aretz the sifter and strainer mentioned above, and she may even do actual work together with her: "U'voreres v'tochenes u'merakedes imah".
The reason for this: as long as the items are dry and have not come into contact with liquid, they cannot become tamei, and therefore tumah is not a factor at that stage. But "mishetatil es hamayim" - from the moment water is added to the dough, it is fit to become tamei, and from that point on she may not touch anything of hers.
The Reasons for the Prohibition and for the Leniency:
"She'ein machazikin y'dei ovrei aveirah" - this is the reason for the prohibition against direct involvement: we cannot support and strengthen those who violate the Torah.
"V'chulan lo amru ela mipnei darchei shalom" - the things that were permitted were permitted only in order to preserve harmony and peaceful relations between people.
"U'machazikin y'dei goyim ba'sheviis":
During the Shemittah year a gentile is permitted to work the field, and therefore one may strengthen his hands - that is, tell him that he should succeed in his work. This does not mean physical assistance, which may not be given, but only words of encouragement. Such words of strengthening and encouragement, however, may not be said to a Jew who works during Shemittah, since he is violating the Torah.
"V'sho'alin bishlomam mipnei darchei shalom":
Since the Mishnah is dealing with offering moral support to gentiles, it adds that one may inquire after their welfare on any day, for the sake of peaceful relations. The Gemara explains that there is no doubt whatsoever that one may inquire after the welfare of gentiles, and that is not the Mishnah's novel teaching. Its novelty is that even on their festival days one greets them in the same way as on any other day, and there is no concern that this constitutes support for their festivals.
In summary: In this Mishnah we learned the limits of assisting others: one may lend utensils to a woman suspected regarding Shemittah, but one may not sort or grind together with her; the wife of a chaver may work together with the wife of an am ha'aretz as long as water has not yet been added, but once it has been added she may not touch anything of hers. The reason for the prohibition is that we do not strengthen the hands of transgressors, and the reason for the leniency is peaceful relations. We also learned that one may strengthen the hands of gentiles during Shemittah but not those of a Jew, and that one inquires after the welfare of gentiles on any day, even on their festival days, for the sake of peaceful relations.