Now, if you had dial up and paid a subscription for access to a free phone number and they cut you off before your subscription ended, then yes you absolutely would expect a refund.
And, regardless, if my ISP stopped providing dial-up service, and I really still wanted dial-up service, I would have simply found another ISP to connect to. (After expecting my previous ISP to issue me a pro-rated refund for whatever subscription I was paying them.)
Also regardless, your analogy is bad, as the modem was a one-time cost. The issue at hand is that of recurring subscription fees Twitterific charges. If we were just talking about a one-time cost to buy the Twitterific app, I wouldn't be suggesting the refund that cost (except to customers who just bought the app right before the shutdown, perhaps; seems shitty to keep customer money when the product irreparably breaks a few days after purchase).
So, no, if Netflix disappeared in this case, I would not ask for a partial refund. The TV still works as a TV, and that's what I really paid for in the first place. (If Netflix didn't disappear, but the TV app stopped working, I might be -- justifiably -- annoyed, though!)
This Twitterific thing is in no way comparable: the app only interacts with Twitter. That's its entire reason for being. If it can't do that, then it is worthless, and anyone who paid for it is perfectly in their rights to demand a refund. Hell, the Twitterific folks should be pro-actively refunding the unused portions of subscriptions; that's the only ethical thing to do here.