Q: May the Extraordinary Ministers place the consecrated hosts not consumed at Mass in the Tabernacle?
A: I treated this in my book, Let Us Pray: A Guide to the Rubrics of Sunday Mass.
While treating how the priest celebrates mass without a deacon, GIRM 163 instructs him to consume what remains of the hosts or place them in the tabernacle. There is no treatment for what to do when there is a deacon or extraordinary ministers. Especially if the tabernacle is in a space apart from the sanctuary, it makes more sense for another minister to place the unconsumed hosts in the tabernacle. And that is my current practice. However, in my previous parish, where the tabernacle was quite close to the altar, it made more sense for me to put the hosts away while the extraordinary ministers assisted with other duties.