The Tur in Even HaEzer brings the shitta of the Sefer HaIttur that if someone is amichalel Shabbos in public, then his kiddushin is not a kiddushin since he has the status of a non Jew. The m'kor for the Ittur is a possuk in this week's parsha "beini u'ben bnei yisroel os hi l'olam". If you don't keep Shabbos then you are not part of Yisroel.
There are 2 interesting shailos discussed in the poskim that are relevant to us. One is whether one can drink the wine touched by a michaleil Shabbos and the other shailah is whether one can cook on Yom Tov for a michaleil Shabbos (just like you can't cook for a non Jew on Yom Tov, you would not be able to cook for a michaleil Shabbos.
It should be noted that this whole discussion could be moot. That is because there is a shitta of the Chazon Ish that most non frum Jews today are in the catagory of a tinok she'nishba and would not be considered a michaleil SHabbos b'pharhesia. The Chazon Ish (Yora Deia Siman 1:6) writes that if someone would invest the appropriate hishtadlus necessary to convince a person to be frum, and that person would become frum then as long as there was no appropriate hishtadlus done, that non frum person is a tinok she'nishba. According to this Chazon Ish it would appear that the only question to be asked is what is called appropriate hishtadlus and for each person the answer would be different.
Status of Wine
Rav Tzvi Pesach Frank in the Har Tzvi discusses the status of the wine. He writes that it is clear from a number of places (Teshuvos HaRashba, Rivash) that the wine of a michaleil Shabbos b'pharheisia is assur. He says some want to argue that the whole reason the wine of a non Jew who does not worship avoda zara is assur is because we are afraid if you drink their wine you will marry their daughters (mishum chasnus). However, there is no issur to marry teh daughter of a michaleil Shabbos. Therefore the wine should be muttar. The Har Tzvi rejects this and says once the gezeirah was made not to drink wine of a non Jew, we don't look at the reason. If a michaleil Shabbos has the din of a non Jew then his wine is assur even if his daughter would be muttar.
It seems that the Chazon Ish disagrees. In Yora Deia (2:28) he writes that since the reason of chasnus does not apply to a michaleil Shabbos, his wine is muttar. Furthermore, the Rambam doesn't mention it is assur and even the Rashba who holds it is assur doesn't bring a m'kor.
The L'horos Nosson (chelek 1 siman 39) and Tzitz Eliezer (chelek 8) also discuss this in relation to having non frum waiters at hotels. Are they allowed to serve wine? The L'horos Nosson mentions the sevara of the Chazon Ish but he says that the Rishonim are mefurash against the Chazon Ish. The Ramban in Chullin says that even though the daughters of a mumar is muttar. He also brings the Chasam Sofer that the wine is assur as a k'nas and not because of chasnus. Furthermore, the Chazon Ish himself in Yora Deia (49:7) paskens the wine of a michaleil shabbos is assur because of a knas, even though there is no issur chasnus.In the end, the L'horos Nosson is meikel for several reasons a) they have a din of tinok shenishba like the Chazon Ish b) they would be embarrased to be m'chaleil shabbos in front of a talmid chacham and therefore it is not b'pharhesia c) the waiters don't actually touch the wine with their hands. The Tzitz Eliezer vehemently disagrees and he says they are not tinok shenishba and the heter of not touching the wine does not apply here. He also quotes the sevara of the Har Tzvi as well.
Cooking on Yom Tov
Interestingly, while the Tzitz Eliezer is machmir by wine, he is meikil by cooking on Yom Tov. He writes that the only reason one can not cook for a non Jew is if you don't use the sevara of "Ho'il"-this sevara says I can cook on Yom Tov even though noone needs the food because maybe guests will come. However,m we do say this sevara. There is a tzad not to use "ho'il" beacause maybe you will cook non kosher food for teh non Jew and you coudn't argue you are cooking to feed guests. However, by a non frum Jew this chashash does not apply. At the end of the day he allows you to cook for a non frum Jew.