Moving postresql 9.1 database - live
If the PostgreSQL version and architecture is the same on the target so they're binary compatible: Set up streaming replication with WAL archive fallback, so the new host replicates from the old one. Enable hot_standby mode so you can connect to the new host read-only. Test. Then when you're ready to cut over, stop the old server and pg_ctl promote the new server to a read/write master.

If the PostgreSQL version is different on the target: Set up Londiste or Slony-I logical replication from the old DB to the new one. Let it sync up. When ready to switch over, shut down the old server, stop the replication service, switch clients to write to the new server.