I'll get the conversation started with my personal view: browsers, at least in their current form, belong on computers, tablets and smart phones. Other appliances do not need browsers.
One major example is the TV. It is clear by now that people don't want the Internet in their TVs. Refrigerators with Internet connectivity have failed to catch on, and we can easily go on.
The main reasons are usability and convenience. Once we can interact naturally with a browser (i.e. not necessarily using a keyboard or a mouse), browsers might be acceptable on TVs. You'll be able to ask things like "what's this actor's name?" or "replay that commercial, freeze that frame, go to that web site" and get a response from your TV. Even then, if you are watching TV with others, it will get pretty annoying if everyone starts interrupting the show to go off on a web site for each commercial. It might be easier to say "remind me about this product" and then get a link on your PC/table/phone. You'd still be browsing elsewhere for the most part.
The way the Internet will become ubiquitous is through smart homes and appliances. They will become as smart as they need to be, but not smarter than that. I may want to check the family's schedule, or the weather on the fridge but I wouldn't want to stand and read a news article there. I would like to be able to scan the bar code on an empty box of milk or eggs using the fridge door and then get a reminder on my phone on the way home that I need to pick up that product. I would not want to answer a question on Quora standing in front of the fridge, though.
Finally, one big reason other appliances won't need browsers is that our tablets and phones are powerful enough to run full-blown browsers that we carry in our pockets or hands all the time. That really reduces the need for full browsers in other appliances