We have a very “chashuv” neitz minyan in Johannesburg. On Yom Tov there is one person who doesnt keep Shabbos, who has been a member of a Shomer Shabbos Shul for over 15 years, who comes to daven at the neitz minyan so he can get to work early and open his shop.
He has been told in a nice way to start keeping Shabbos, but still doesn’t. The question is, can this man, who is a Kohen, Duchen? Can he get an Aliyah (even Gelilah), and can you count him for a Minyan?
This is a delicate question, and depends on circumstances.
It is generally permitted to give the person an aliyah (where necessary), and he should is not prevented from ascending for the Duchen, but he should preferably not be counted for a minyan.
He should continually be told that he should keep Shabbos, and the more pressure that the community can apply on him, the better.
Please see below for sources and explanations.
The Chacham Zvi (38) writes with great force that somebody who desecrates the Shabbos publicly, “who is considered as a non-Jew” (his words), may not be given an aliyah, “for how is it possible that he will ascend to read from the Torah in public, for there is no greater chilul Hashem and extinguishing of religion than this, and anybody who is present and does not object will be ensnared in his sin.”
Yet, this was true for his day, and not necessarily for ours. Today, permitting those who violate Shabbos to receive an aliyah is not (in general) a chilul Hashem and an “extinguishing of religion,” but quite the contrary: By allowing such people to receive an aliyah, we keep them in the fold, and allow the a chance to repent their ways.
It is certainly with this in mind that Rav Moshe Feinstein (3:12) writes that with the exception of actual kofrim, those who are known to be avaryanim, but do so for reasons of parnassah and so on, can be given aliyos. Note, however, that it Vol. 4 no. 91, sec. 8, the Iggros Moshe writes that those who violate the Shabbos publicly should not be given an aliyah.
Orchos Rabbeinu (p. 122) writes, in the name of the Steipler, that one can give an aliyah to somebody who violates Shabbos, because if he comes to shul he demonstrates that he still have a connection and a feeling for religion, and it is therefore permitted to call him up.