Many security risks have been posted about Smart TVs. Some included browser based attacks, some allowed attackers to upload malware, but at the end of the day it is not the Smart TV that is the problem, it is what the TV is RUNNING as an operating system that is the problem.
I don't believe there is a catch all answer, it will be dependent on what TV you purchase. Let's have a look at "Android TV" which is used by Sony's Bravia. The versions of Android used are constantly under attack.
So what can you do with Android TV? Well you can check your Facebook, Twitter, and other social media accounts which means that at some point in time, credentials are being taken from you, processed on your TV, then sent to those sites. Browsing? Well that can lead to purchasing something via Credit Cards, etc. There would be "storage" concerns (credentials, pictures, cache), leveraging your TV to perhaps perform attacks against other computers, connection concerns - are they done via SSL? How does my TV's browser store certificates, can someone upload malicious certificates?
The answers to me would align with any normal computer, and although there is little information available about the security behind Smart TVs, the likelihood of someone stumbling upon your TV on the Internet is highly unlikely if you are at home. You would need to FORCE port forwarding/DMZ/etc. This does not include any potential client side (browser) attacks, but my best guesstimate is, the attacks are similar to that of a system being used to do daily browsing, etc.