The short answer is yes, Christians should be obeying the Old Testament Law (although I would take issue with calling it "old testament" in the first place, preferring instead to call it Torah, God's Teaching). To be sure, didn't Paul and the rest of the "New Testament" Jewish followers of Yeshua (Jesus) continue to obey Torah post resurrection? But that was 1st century. However, in the 21st century, much of the Torah cannot be kept because we have no functioning Temple, no priests, no animal sacrifices, and no theocracy ruing over the Land of Israel. What is more, keeping the Torah in the Land of Israel has to be modified when one dwells outside the Land--even if there was a standing Temple.
It is my understanding that the errors surrounding one’s relationship to Torah can be corrected once a person resolves the issues surrounding identity and legalism, begins to understand the intended nature and function of the Torah in the first place, and then faithfully applies it to their own lives. Because the Messiah has already come, the Torah is now a document meant to be lived out in the life of a faithful follower of Yeshua, through the power of the Ruach HaKodesh (Holy Spirit), to the glory of HaShem (God) the Father. It should not be presumed that it could be obeyed mechanically, automatically, legalistically, without having faith, without having trust in HaShem, without having love for HaShem or man, and without being empowered by the Ruach HaKodesh. To state it succinctly, Torah observance is a matter of the heart, always has been, and always will be.
Because the Torah is written on the hearts of all who truly name the name of Yeshua as LORD and Savior, it is meant to be followed to the best of our ability. We have no reason for fear of condemnation, or the trappings of legalism!
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