|You don't need the Internet to use a router on a LAN. Most routers have built in Swtich that connects to the LAN. The Wireless part is on the LAN side of the router.|
However, it probably wont just work out of the box. You are going to need access the the routers setup pages in order to configure the wireless to the same protocols the iPhone is using or configure the Iphones to use the default protocols of the router which in most case means no security. That means a computer with an Ethernet connection. Very few wireless routers will work out of the box and a cheap Linksys router is unlikely to be one of them.
Once the router is set up and configured properly there is no reason why the iPhones should not be able to communicate through it.