Samsung showed off their Tizen TV Prototype to developers at the Tizen Developer Conference today. The user interface is a work in progress at the moment, but does show what the TV is currently capable of, and will have the ability of integrating further services in the future.
Tizen is a true linux open source operating system, which brings with it stability and speed. The Tizen TV should be able to side load standard Tizen tpk and widget applications. There has been no formal decision on product features, and this being a prototype TV it is not expected to be feature rich at this point.
Once you mention standard linux applications, the stituation is a bit more complex. Not having root access, and moving to wayland in the future introduces its own complexities. If the applications are written portably without dropping down to lower API's, there is no reason for them not to run.