The gemara tells that R. Eliezer was captured and almost killed and was reminded by R. Akiva that it was a punishment for the time where he appreciated a p'shat in a pasuk that was told to him by a "min". The punishment was for violating the pasuk in Mishlei that warns to distance oneself from "minus" - those who reject Hashem. Tosafos points out that although we find that various amora'im would frequent "bei avidan" to debate the priests, that was not a house of worship rather a place of debate and it was therefore permitted to enter.
It is well known that a Jew is not allowed to enter a church. Where does this come from? R. Moshe (Igros Moshe Y.D. 3:129:6) understands that this gemara is the source of the prohibition. However, there seems to be a more explicit source for this issur. The mishna earlier (avoda zara 11b) says that one is not even allowed to take a road to a city of avoda zara, unless the road also leads somewhere else. Rashi explains that the prohibition is because it looks like he is going to worship the avoda zara. The Rambam in his pirush ha'mishna elaborates:
ולפיכך יודע לך, שכל עיר של אומה שיהיה להם בה בית תיפלה שהוא בית עבודת כוכבים, בלא ספק אותה העיר אסור לעבור בה בכוונה וכל שכן לדור בה. אבל אנחנו תחת ידיהם בעונותינו ושוכנים בארצם אנוסים ונתקיים בנו מה שנאמר "ועבדתם שם אלהים מעשה ידיד אדם עץ ואבן" , ואם העיר דינה כן, ק"ו דין בית אבודת כוכבים עצמו שהוא אסור לנו כמעט לראותו וכל שכן ;ליכנס בו
The Rambam holds that this mishna is the source to prohibit entry into any house of worship of other religions. The Rambam says that it is "almost" prohibited to even look at these places, never mind entering them. It is well known that the Rambam regarded christianity as idoaltry (unlike tosafos 2b who writes that they goyim we live among are not idol worshipers). Even Tosafos may hold that christianity is avoda zara, just that we aren't machmir for all restrictions because the goyim we live among aren't so religious. Based on this it would clearly be assur to enter a church. Even entry into a mosque (or Islamic YMCA - :), although not strictly avoda zara according to the rambam, would at least be assur becasue it is "minus" since their system of belief and hierarchy of profits is against the Jewish tradition and undermines all Torah/Rabbinic law. Additionally, Tosafos 17b writes that one should even distance himself from the opening of a house of avoda zara. Meaning, that if there are two paths to take, one of which opens to avoda zara, one is obligated to choose the other, as the amora'im would have done (if not for the fact that the other was open to a beis zonos).
It is also interesting to see the Shiltei Ha'giborim (pages of ri"f 5b) citing the Ria"z who elaborates about the prohibition to study philosophy, particularly Aristotle and his buddies who were all kofrim as a violation of distancing from "minus". He continues to explain that one need not prove G-d's existence, nor must he be able to philosophize about G-d. He advocates אמונה פשוטה, simply believing because that is our traditio - וכן צוה עלינו רבן של נביאים ע"ה שמע ישראל ה' אלקינו השם אחד, ולא אמר להשכיל ולהתבונן בידיעת האלקים בדרך החכמה אלא להאמין היחוד על פי השמועה ועל פי הקבלה כמו שאנו מקובלין כל התורה והמצות. The Rambam clearly held differently and spent much time on the study of philosophy. There is much discussion about the translation of the Rambam's sefer hamitzvos in his first mizvah whether the mitzvah is to "believe" (which would imply blindly) or to "know" (which would imply through study). See footnote of R. Chaim Heller here.The translation of "know" - לידע, is far more consistent with the Rambam's opening of mishna torah - לידע שיש שם מצוי ראשון. The Rambam spent much time on developing belief in G-d through philosophy and considered the mitzvah of belief in Hashem to be ongoing - as one learns more they appreciate the existence of G-d more than before. The Rambam held that the study of G-d's existence is essential and demanded by the first mitzvah in the Torah, completely rejecting the notion that of אמונה פשוטה.