What an excellent question! They were many rabbis who supported and opposed the Rambam's 13 principles of faith.
In his The Limits of Orthodox Theology: Maimonides’ Thirteen Principles Reappraised, Rabbi Marc B. Shapiro examines the principals. In his Commentary on the Mishnah, Introduction to Perek Chelek, Maimonides calls this dogma; whoever rejects these principles loses their share in the World to Come. Menachem Kellner calls the principles the first dogma of Judaism. But some have condemned this as a scare tactic, similar to those used in Christianity (accept Jesus or else). Yet, as we will see, even well-respected Orthodox rabbis rejected his ideas. Some speculate on whether Rambam himself accepted them. We will go through a few (but of course not all).
13 Principles reappraised
There is many responsa literature on the subject. While Maimonides listed 13, Shimon ben Zemah Duran, in his Ohev mishpat and Joseph Albo, in Sefer Ha-ikkarim listed only three. Hasdai Crescas, in Or ha-shem listed only six. David ben Yom Tov ibn Bilia lists twenty-six principles of faith. Don Isaac Abravanel lists 613.
The Tosaphist Rabbi Moses Taku disagreed with Maimonides' first principle. He felt that G-d could do the impossible (create a rock so big that He cannot lift and yet lifts it). He also felt that sometimes G-d can take human form and calls Rambam's position that G-d does not change and is always incorporeal, thus denying that G-d sometimes change is considered heresy.
Rabbi Abraham ben David of Posquieres (known as Rabad, or Ra’avad,) disagreed with the second and third principles'. Writing, "There are many people greater and superior to him who adhere to such a belief [that G-d has a body like humans] on the basis of what they have seen in verses of Scripture and even in the words of those aggadot which corrupt right opinion about religious matters.” He also wrote, “greater and better people than Rambam” were corporealists. Comment to Hilchos Teshuvah 3:7. According to Rabbi Slifkin Rashi was a corporealist. You can read that article here. Additionally, Kabbalist believes that G-d is made up of ten parts called Sefirot.
Abraham ibn Ezra rejects the fourth, saying, that G-d created the world out of preexisting matter. Some scholars dispute whether the Rambam might have also felt this way and might have used "Nessecary beliefs" to state the opposite if that was not his true conviction. (more on that later).
Some Jews seemingly reject the fifth principle when they pray to angels every Friday night in the Shalom Aleichem, which states: “Bless me for peace, angels of peace.” Others have rationalized it, saying that it's not to be taken littery. Maimonides himself seemingly rejects the sixth principle, stating that prophecy is the result of a higher intellect.
The Babylonian Talmud seems to disagree with the 7th. Sanhedrin 21b states that Ezra is greater than Moses: “If Moses had not preceded him, Ezra would have been worthy to have the Torah revealed to Israel through him.” The Midrash Genesis Rabbah 14:34 thinks it was Balaam: “What prophet did they [non-Israelites] have who was like Moses? Balaam the son of Beor.” Regarding the 8th, the Masoretic text has changed a few letters, though with minor variations. See the Midrash Numbers Rabbah and Avot d’Rabbi Natan regarding the Masorites placing dots.
Regarding the ninth, the Babylonian Talmud, Nidah 61b, Rabbi Joseph says: “The mitzvot [commandments] will be abolished in the time to come.” Rabbi Joseph Albo, in his Sefer Ha-ikkarim) writes that a future prophet can abolish [or, nullify] all of the biblical commands except the Decalogue (the Ten Commandments). Regarding the 10th, ibn Ezra writes in Genesis 18:21, “The whole [G-d] knows the individual in a general manner rather than a detailed manner.” Ralbag writes similarly in Milhamot Hashem, that G-d knows the universal but not the particular. Many think the Rambam disputes reward and punishment in his Commentary on the Mishnah, Introduction to Perek Chelek. Rabbi Hillel (Babylonian Talmud, Sanhedrin 99a), would reject principle 12, stating that the messiah already came in the days of King Hezekiah. Regarding the 13th, Rabad writes in his Commentary on the Mishnah, Introduction to Perek Chelek and Mishneh Torah, Hilkhot Teshuvah 8: “There is no resurrection for bodies, but only for souls.” Whether or not bodies resurrect spiritually or physically, all agree that Maimonides wrote that after the messianic age no bodies exist, only intellects.
Some scholars think Maimonides himself did not believe in all of his principles, only the first few that talks about G-d. Some scholars have pointed to a concept called "Necessary beliefs." Spinoza, as well as many Arab philosophers, such as Averroes, used this tradition called Plato's Noble Lie. It was sometimes used to keep civilization in check, to tell untruths or essential beliefs because the masses couldn't grasp deeper philosophy.
In short, many well-respected rabbis accepted or reject these views. Shapiro stresses in his book that what matters more is behavior. He puts emphasis on action over beliefs.
Personally, I think Maimonides accepted all of his principles. These [above] are only some of the views proposed by Rabbi Marc B. Shapiro in his book The Limits of Orthodox Theology.