If I am visiting a community that has an Eiruv, and I do not want to (shouldn’t?) use it, can I ask someone to carry an object for me (to/from Shul for example)?
Firstly without knowing the Halachic status of the Eiruv it is difficult to answer. If you are referring to a metropolitan city Eiruv those issues are even more questionable, and as we know the Rebbe was generally not supportive of the construction of an Eiruv in a metropolitan city unless done secretly.
In general, if you are of the opinion or understanding that the Eiruv is not kosher you cannot tell another Jew to carry on your behalf. This is because he is considered your Shaliach, acting on your behalf. Additionally, by asking him to do something you consider forbidden, you are enabling him to do the the wrong thing (placing a stumbling block before the blind). This can also appear that one is degrading the Torah observance of one’s friend.
However, if your behaviour stems from a Chumra, i.e. you are acting extra stringent, and not as an Halachic obligation, you may ask another Jew to use the Eiruv (or any other Shabbos Chumra) on your behalf.
See a similar discussion here.
See however Igros Moshe (Orach Chayim 4:119;5) who writes that if one opens a can on Shabbos based on his Rav’s view on behalf of another Jew who follows the view that it is forbidden to open. The second Jew may eat the food on Shabbos since the opening was done permissibly. However, there it is not talking of the second Jew telling the first Jew to open on his behalf, rather that after the fact that it was already opened the second Jew may be lenient.