Are you allowed to call someone to the torah who does not keep Shabbat?
Common practice is to be lenient to give an aliyah to someone who is not scrupulous in Sabbath observance. For one who is totally and openly not Observant generally aliyot are not given, unless it is a means of bringing them closer to Judaism.
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 often 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.