It's not just ITV Be, all ITV channels are streamed live on the Roku box.
However, what you are asking does present an interesting question.
Traditionally, any TV that could pick up ITV, could pick up the BBC as well. As the TV license only financially supports the BBC (none of the money goes to other networks), it still didn't matter if people never "tuned into" the BBC as the TV could technically still be tuned at another state.
But the Roku player can not "tune into" the BBC at all - under the standard apps from the app store (unless I don't know of an app that can). So the argument could still be that the Roku player doesn't qualify as a proper television (presuming the monitor is not a normal TV).
Oh... a slight problem with this is that FilmOn have a "private" Roku app which does indeed allow live BBC, so that could throw a spanner into that idea...
Still, I really don't understand why some people try so hard not to pay the TV license. The whole reason such a tiny country like the UK stands up so high around the world for the output of their television industry is because of the TV license and the BBC itself.
I know some people feel because they don't watch it, then why should they pay for it, but that argument can be asked about tax in general... I never needed (yet) the fire department, so why should my taxes pay for their upkeep? (some parts of the US are like this, where if you don't pay private fire insurance, the fire department will not put out your house fire and let your home burn to the ground). Or, because I don't have kids, why should my taxes pay for schools? I don't drive, why should my taxes pay for roads etc.
The BBC license fee is like this, but better than a tax - because if it were a tax, then the incumbent government could get their dirty little fingers in and turn it into what their political ideologies would like.
Anyway...