If you want to use the online functions of your viera directly than yes you will need an adapter. It seems like more and more of our home theatre hardware is network capable nowadays. Not only for streaming video, and online gaming, but also for basic firmware updates.
For this reason I usually recommend moving the router and modem to the tv stand. This way all of your speed hungry video streaming and online gaming needs can be fulfilled with a solid hardwired connection. Then you just need to look in to one of these for getting internet from your modem (in your entertainment system in the living room) to your computer. It works better than wireless by actually sending the ethernet signal through the electrical wiring in your house. 
. It is also about $40 cheaper than most proprietary adapters needed for tv's. Of coarse you can go the opposite route by leaving your modem and router with your computer, and using the netgear powerline kit to bring the internet to your home theatre, however email, facebook, online banking, etc will need far less bandwidth than Call of Duty or netflix.
