It is not possible to stream BYUtv to the wii. The wii's Opera browser uses an out of date version of flash that Nintendo hasn't bothered to update in a long while and considering the fact that they are pressing forward with their new Wii U system so aggresively, I doubt there will be any more updates in the pipe. As such, the wii browser only supports Flash 8 and parts of Flash 9, both of which are out of date. It is technically possible that someone could program a homebrew application that would remedy this, but I am not aware of any that exist currently and I doubt there is much developer interest in this area. The wii is quite crippled in terms of RAM that is available to the browser, and flash is very RAM heavy, so I doubt anybody would really want to try.
As for iphones, you're in much better luck. You should be able to pick up a cable that you can just plug into your tv. They run anywhere from 20-50$. You could then use the BYUtv app on the phone and watch it on your tv, but I'm not sure what the quality would be like as the feed is compressed quite a bit for mobile devices. It might be worth a try though. Also, if you have an apple tv, you could easily stream the feed on your phone over wifi to the apple tv. But at 99$, I'm not sure apple tv is worth it just for BYUtv.
The Roku isn't a bad idea at all. If, however, you have a laptop, you could very easily just hook that up to your television with a VGA, DVi, or HDMI cable.