schroot supports LVM and Btrfs snapshots. From its man page:
Some chroot types, for example LVM snapshots and Btrfs snapshots, provide session-managed copy-on-write snapshots of the chroot. These also provide a source chroot to allow easy access to the filesystem used as a source for snapshotting. These are regular chroots as well, just with the snapshotting disabled. For a chroot named "sid-snapshot" (i.e. with a fully qualified name of "chroot:sid-snapshot"), there will also be a corresponding source chroot named "source:sid-snapshot". Earlier versions of schroot provided source chroots with a '-source' suffix.
These are also provided for compatibility. In this example, this would be called "chroot:sid-snapshot-source". These compatibility names will be dropped in a future version, so programs and scripts should switch to using the namespace-qualified names rather than the old suffix.
Otherwise you can look at other copy-on-write filesystems (in short COW), as