The hardware required depends on the size of your implementation. For small implementations, OpenMRS can be run on just about any desktop or laptop.
1 GHz processor or better, 256 MB of memory or more, 40 GB hard drive or larger. You can set up the server on a laptop for demonstration or testing purposes.
1.5+ GHz, 2 GB of memory, and 150+ GB of disk space with RAID and appropriate backup facilities.
AMPATH is supporting an OpenMRS instance with over 250,000 patients with two 2.26 GHz quad processors, 16 GB of memory, and 500 GB of disk space with RAID and appropriate backup facilities.
You can view profiles of real-world implemented systems here.