In parishes, usually the remaining Hosts are in a ciborium, which is placed in the tabernacle.
On occasion I go to Mass at our nearby Trappist abbey on a Sunday. There, the remaining Hosts are consumed by the monks. Before Mass, those who wish to receive will place an unconsecrated host on a tray, which is brought up after the homily; the monks have one or two large hosts which are thicker than a normal host, and those are cubed and primarily received by the monks. After Communion, there may be a few consecrated Hosts as noted and are distributed to several monks and consumed.
They have a tabernacle, and will keep a minimum number of consecrated Hosts in it should some be needed outside Mass, but they retain only a minimum, so those left over after Communion are not put into the tabernacle, but rather consumed.