Question: My neighbour doesn’t use the eruv though sometimes asks me to carry things on his behalf. Is that allowed?
Answer: The Shulchan Aruch (OC 263:17) writes that one who accepted Shabbos early may ask another who hasn’t yet accepted Shabbos to perform a melacha on their behalf (See Tosafos, Shabbos 151a).
Nonetheless, R’ Eliezer Waldenberg (Tzitz Eliezer 18:32:3) differentiates between this scenario and asking another to perform a melacha when it’s Shabbos for him, too. Thus, a sefardi who follows a stricter position mustn’t ask an ashkenazi who follows a more lenient position to perform a melacha on his behalf.
The Gemara (Shabbos 150a; Bava Metzia 90a) teaches us that it is assur miderbanan to ask a non-Jewish person to do melacha for them on Shabbos. The Shulchan Aruch (OC 276:1) writes that if one mistakenly did so, they would be forbidden to benefit from that action on Shabbos.
Thus, R’ Moshe Feinstein (Igros Moshe OC 4:119:5) writes that one who doesn’t open cans on Shabbos mustn’t ask a non-Jewish person to open it for them. If a non-Jewish person opened it for them, they wouldn’t be allowed to eat that food on Shabbos. Nonetheless, if another Jew opened it for him as they follow poskim who allow doing so (See Minchas Shlomo 2:12, Shemiras Shabbos Kehilchasa 9:n10; Yechave Daas 2:42), then they may eat from the can. Similarly, the Mishna Berura (318:2; 27) and R’ Waldenberg (ibid.) allows one to benefit from any such melacha performed.
Based on this, R’ Ribiat (The 39 Melochos, p93) writes that one who doesn’t open bottles or cans on Shabbos, should not ask those who do, to do so for them.In conclusion, one who doesn’t use the eruv on Shabbos should not ask you to carry on their behalf. They may benefit from you doing so, however.