I have a iPad 3 or the new iPad and it's 3G. I plan on using this at trade shows so I can mirror what's on my iPad to the TV using Apple TV. We'll be using a local router instead of wi-fi because it's a fortune to purchase internet at shows. My question is once I access an app that requires 3g connection will I lose my connection to Apple TV?
As far as I know the iPad can only have one IP-based connection at a time. That is either local Wifi or 3G. I am not exactly sure if you will loose the AppleTV connection or you won't be able to access the internet, but I sure you won't be able to have both.
You might want to try to bring another iPhone or iPad and use it's Personal Hotspot Feature. You would then connect the AppleTV and your new iPad to this Personal Hotspot Wifi.
Please note that the 3G is usually very crowded at trade shows, because you are not the only one using it ;-)