When Moshiach comes, will we continue to observe all the Jewish laws that are kept today?

As a rule, our sages say: "The laws of the oral Torah will never be nullified." However, there may be a few modifications to those laws. The most fundamental will be the resolution of the age-old debate between the schools of Hillel and Shammai. 

Hillel and Shammai were the leading two sages of their time, and they disagreed over many interpretations of law.  The school of Hillel had more disciples than the school of Shammai, and since Jewish rulings follow the majority, the law was decided according to the school of Hillel.  However, in the future the majority of scholars will agree to Shammai's interpretation, so the halacha will follow his ruling.

What will change? Rabbi Elchanan Wasserman explains: "The school of Shammai was sharper than the school of Hillel, and in truth their interpretation was correct.  However, most students were not capable of following the depth of their reasoning, and the halacha was therefore decided according to Hillel. In the future, though, when the world will be filled with knowledge of G-d, it will become clear that the halacha is according to Shammai.

Rabbi Shmelka of Nikolsburg offered a similar explanation: "Today, the halacha is decided according to the school of Hilel, since the level of Shammai was above this world and could not be contained in this universe. However, when the Redeemer comes the ruling will be according to Shammai."

A similar change will happen with regard to the rulings of Rabbi Meir. Wherever there was a dispute between Rabbi Meir and his colleagues, the halacha was decided according to the majority, against Rabbi Meir. However, when Moshiach comes the Torah scholars will finally appreciate the depth of wisdom in Rabbi Meir's rulings, and the halacha will be decided like him.

Talmud Yerushalmi, Megillah 1:5.  Likutei Torah Korach 54:2.  Tosephot Hachadashim Avot.  Divrei Sofrim 5:17.  Midrash Pinchas 1:53.  Torat Menachem 5751, vol. 3, p. 284.  Mikdash Melech Bereishit 17:2