Not necessarily. You need a platform which supports Linux syscalls, but you don't need Linux per se. Unlike VMs, containers sit above the kernel so you're tied to its interface but not its implementation. For example, see Solaris' LX Branded Zones.
At Joyent, our Triton cloud stack is built on illumos (nee Solaris) LX branded zones. We also have a Docker-compatible remote API, so not only can you run unmodified Linux binaries in containers, you can do so using the regular Docker client CLI!
