The simple answer is maybe
The Linksys BEFSR41 is a great little router for home use. For the very limited role and specific functions that it performs, it is reliable and cheap. I've certainly found them to be far more reliable than their fancier wireless counterparts, even with open-source project firmware loaded.
The problem with the function you need to perform is that the BEFSR41's main role is to be a Network Address Translation firewall. Everything on the LAN side runs privately, and it will forward connections that originate on the LAN side to the WAN side, but block every connection that originates on the WAN side.
Prior to v16 of RSLogix 5000, such a discussion would be a non-starter; the UDP Multicast traffic used by I/O connections and Produced/Consumed Tags won't cross any kind of router because the Time To Live (TTL) parameter is hard set =1, and every router decrements that by 1 when it passes a packet.
v16 of RSLogix 5000 allows an option to perform Produced/Consumed tags between just two devices (you cannot have multiple consumers) using a Unicast target address. I believe such a packet can cross a router.
v18 of RSLogix 5000 provides similar functionality for single-owner I/O connections.
The BEFSR41 might work with Unicast Produced/Consumed Messages if the Consumer is on the LAN side. The Consumer's requests will go through the router to the WAN side, and replies to those requests will come back. UDP Unicast packets produced by the Producer *should* also pass through the router.
The BEFSR41 also might work with Unicast Produced/Consumed Messages if the Consumer is on the WAN side and you configure the IP address of the controller on the LAN side to be in the "DMZ", or if you configure port forwarding to send Port 44818 only to the IP address of the controller on the LAN side.
That's two "mights" and a "should". Not something I'd strut confidently onto the factory floor with, but in my opinion it can't hurt to try, since the small budget is their idea.
My principal concern is not the DMZ or the Port Forwarding or the Multicast filtering, but rather any possibility that the router will attempt to perform some kind of inspection on the Produced/Consumed packets flowing from side to side of the network. If there is any kind of antivirus firewall or "stateful packet inspection" going on, the router will probably choke in a hurry on thousands of tiny packets a second. One of my colleagues broke a BEFSR41 irrevocably on a test bench when he turned off IGMP snooping and the router attempted to analyze a few thousand I/O packets per second. It wouldn't even hard-reset... he had to throw it out.