There could be a sleep mode feature that is causing your router to drop its connection with the printer. Normally a small heartbeat will pulse out from the printer periodically to let the network know it is still alive but in sleep mode. Your router may not be understanding those signals or the printer may not actually be sending them.
A couple of things you can try:
1. Eliminate WiFi as the connection between pritner and router. Connect a normal Ethernet cable for a while and see if the same sleep mode issue is consistent or not. This will help you isolate the problem to a WiFi or firmware issue on the printer.
2. Rig up a small ping script. There are many ways to accomplish this but in a nutshell you need a small application to go out and send a ping/heartbeat signal to the pritner every xx seconds. Figrure out if you can keep the printer on the network if you force it to communciate with a computer on a regular basis or not.
3. Update your firmware on your router and printer. Should be obvious but wasnt mentioned in your post. Always make sure everything is as updated as can be when troubleshooting printer problems.
Post back and let us know if you can discover anything new or not.