For follow-up discussion and general commentary on the topic. Comments are sorted chronologically.
Ezekiel Kimosop - Pastor & Bible Scholar
This is a difficult question but I agree with Michael that the animal sacrifices will only serve symbolic instruments of Temple worship and cannot in any way take the place of the atoning blood of Christ. However, it should be noted that this religious practice is most likely to be done by Jews who still observe the Mosaic laws and have no bearing on the Gentiles.
Some scholars have argued that this prophecy is merely symbolic and that no real sacrifices will be literally reinstated by Christ who by his death had put them to an end.
Ariel HaNaviy
We KNOW that ritual impurity was a reality in the times of the TaNaKH (OT). Anyone reading the book of Leviticus will notice this truth. So, how did those Jews and Gentiles in ancient Isra'el return to a state of ritually purity? Easy. Most often, the substitutionary feature of animal sacrifice provided that function. Their ritual impurity was not merely "symbolic." Hebrews 9:13, 14 explicitly teaches cleansing:
“For if the blood of goats and bulls, and the sprinkling of defiled persons with the ashes of a heifer, sanctify for the purification of the flesh, how much more...”
Commentators will notice that this verse is structured as an “a fortiori” argument, meaning logic that is used to express a conclusion for which there is stronger evidence than for a previously accepted one. In order for the verse to be proving that Christ purifies our conscience, the previous clause about purification of the flesh HAS to be true, otherwise a logical fallacy takes place. The writer is emphasizing that Christ’s inner purification is BETTER (“how much more…”). Take note of the phrase “sanctify for the purification of the flesh” again. This is in agreement with statements all over the book of Leviticus that teach that approaching worshippers will not have their sins simply covered, they will be atoned for on the earthly level.
By the way, many Jews of today (myself excluded) also repudiate the possibility of returning to animal sacrifices. This is not just a “Christian” argument.