Is an Ashkenazi Jew allowed to ask a Sepharadi Jew to do labor for him on Shabbat that he himself is not allowed to do but the Sepharadi Jew is allowed, and vice versa?
Some examples where there are differences related to Shabbat between Jews of different heritage include whether one may open sealed bottles on Shabbat, or whether one follows the opinion of Rabbenu Tam for the time of sunset and when bein ha'shemashot (twilight) begins.
The Shulhan Aruch in סימן רס"ג סעיף י"ז says that one who accepts the Shabbat before dark is permitted to instruct his fellow Jew (who has not yet accepted Shabbat) to do labor for him. The Rama adds that it is permissible to benefit from this labor on Shabbat, and even more so on Motzei Shabbat. And if one prays late on Motzei Shabbat, or finishes his meal late into the night, he may instruct his fellow Jew who has already prayed the evening prayer and said Havdalah to distinguish the Shabbat from the weekdays to do labor for him.
The Shulchan Aruch follows the opinion of the Rashba, but the opinion of the Ran is to disallow a person who accepted Shabbat from asking one who did not yet accept Shabbat to perform labor on his behalf.
There are two different ways to explain the Rashba's opinion:
1) The Beit Yosef explains that it is permissible to ask a fellow Jew to do work on Friday even if for himself it is Shabbat already, because technically speaking if he would not have accepted Shabbat early then he would have been allowed to do that work. Furthermore, in a pressing situation he is allowed to nullify his Tosefet Shabbat.
2) The fact is that for his fellow Jew it is still a weekday, therefore there is no prohibition of asking him to do any work (ט"ז).
With regard to our question, based on the explanation of the Beit Yosef above one would not be allowed to ask a fellow Jew of different heritage to perform any labor on Shabbat that is forbidden for himself, since there is never a situation where it would be permissible for him. He does not have the option to choose not to accept that prohibition as in the case of one who accepted Shabbat early, and so the reasoning of the Beit Yosef to allow it would not apply.
However, based on the reasoning of the ט"ז that as long as it is permissible for the other person one may ask him to do it, then there would be no difference between our question and the example above from the Shulhan Aruch. In both cases for one it is permissible and for the other it is forbidden, and therefore one may ask the one for whom it is permissible to perform that labor on your behalf.
Yet we can still say that there is a profound distinction between our question and the example from the Shulhan Aruch. In the example of the Shulhan Aruch, once a person accepted Shabbat all agree that it is Shabbat for him. Not because he is following a certain opinion, but because that is the law according to all opinions. Whereas in our question, we are talking about two groups coming from two different countries, and each group accepted to follow their own Rabbi. The act we are discussing is not totally forbidden because there exists a different group that says it is permissible based on their custom and tradition.
Either way there is room to be lenient. However, if one learned the relevant topic and arrived at the conclusion that it is definitively forbidden, then he cannot ask his friend to do that labor on his behalf.