The issue is as follows. In last week's parsha (קדושים), we have the issur of לפני עור. When you give a non-religious Jew food he is going to eat it without saying a beracha, thereby violating an issur d'rabbanan. Given this, the question comes up, maybe it should be a violation of לפני עור to give him the food.
The gemara in Avoda Zara (6b) learns out from the pasuk of לפני עור that you are not allowed to give a Nazir wine. The gemara concludes that this is only in a case of תרי עברא דנהרא (literally 2 sides of the river), meaning he cannot do the issur without you, but in a case of חד עברא דנהרא (literally 1 side of the river), where he could do the issur without you (he could ask his neighbor for wine) it is permitted. Tosafos paskens like this, that whenever he can do the aveira without you it is permitted and there is no לפני עור. The Ran there comments that although there is no prohibition of לפני עור, it is still prohibited m'drabbanan because of מסייע לעוברי עבירה, you are helping him do an aveira.
The Shulchan Aruch in Yoreh Deah Siman קנ"א paskens like the Ran, that even if he can do the issur without you it is prohibited. The Rama there comments that י"א like Tosafos that it is permitted וכן נוהגים. However, many of the acharonim disagree with the Rama (Magen Avraham, Shach, Gra) and pasken like the Ran because Tosafos in other places assumes that there is an issur d'rabbanan in such a situation.
The Shach as explained by the דגול מרבבה has a very important קולא. The Shach says that there is no machlokes between Tosafos and the Ran. Tosafos is talking about a case where he is doing the aveira במזיד and therefore there is no issur of helping him do an aveira while the Ran is talking about a case of שוגג and therefore there is an issur d'rabbanan to help him.
It comes out according to the Shach that whenever the person is doing the aveira במזיד and/or there is no way for you to stop him, it is mutar to give him the issur if he could do the issur without you.
רעק"א takes a different tack regarding לפני עור. In Yoreh Deah Siman קפ"א סע' ו the Shulchan Aruch writes that it is prohibited for a woman to cut off the payists of a man according to some opinions. רעק"א comments that according to everyone there would be an issur of לפני עור on the woman. He then states the follwing chiddush. If the man could cut his own payists and is going to do it, the woman can cut them for him and will not violate לפני עור. He explains as follows. If the man cuts his own payists, he violates 2 issurim, מקיף and ניקף. If the woman cuts them, he only violates the issur of ניקף and not מקיף. Meaning, that if the woman cuts them she is saving him from an additional aveira and therefore there is no issur לפני עור as her action is saving him from an additional aveira.
It comes out according to רעק"א that if your action reduces the total number or severity of aveiros committed then there is no issur of לפני עור.
R' Moshe (יורה דעה סימן ע"ב) discusses the following case. A religious caterer asked him if he is allowed to cater an affair where there will be mixed dancing. Is there a problem of לפני עור?
R' Moshe answered that it is allowed for 2 reasons:
1. We pasken like the Shach as explained by the דגול מרבבה and therefore in this case where they are doing it במזיד (and they would not listen to you) it would be permitted.
2. This is not לפני עור. The classic case of לפני עור is where you give someone or help someone do an aveira. Here your action is perfectly mutar both for you and the other person. You don't have to worry that he will use it for an aveira. If that was the case then R' Moshe asks how can you sell anything to a non-religious Jew? If you sell him a pot, he will use it for non-kosher food or בשר וחלב. Yet, no one thinks that there is an issur of לפני עור there. The reason is because since your action is perfectly mutar for both you and him there is no לפני עור.
Based on the above we can answer our original question. It is permitted to give a non-religious Jew food for the following reasons:
1. רעק"א - RSZA (Minchas Shlomo Siman ל"ה) was asked this question and was matir for the following reason. He explained that if you don't give him food he will be insulted and he will violate a bigger aveira, לא תשנא את אחיך. Therefore based on רעק"א reasoning (you action saves him from a bigger aveira) it would be permitted.
2. It seems that we pasken like the Shach (at least by issurim d'rabbanan) and therefore here where he is a מזיד it would be mutar. R' Moshe relies on the Shach as does R' Shternbuch.
3. R' Moshe's sevara applies here as well. You are giving him food which he is allowed to eat. You don't have to worry that he will do an aveira with it.
To sum up, there are a number of reasons why the issur of לפני עור would not apply in many of our interctions with non-religious Jews. Each case needs to be looked at individually to see if the reasons apply.