What does ‘ticket to oblivion’ mean? Is this a frequently used phrase?

What "ticket to oblivion" means here — and, no, it's not a time-worn phrase, although it's not exactly fresh, either — is that the person who is chosen to give the response is not going to be remembered. The spectacle of the State of the Union address, and the office of the Presidency of the United States are so much greater and more memorable than ... well, whatever ... that no one will remember or care who gave the response. Everyone can name the first president of the United States, right? OK, who was the first vice-president? Who can even remember who gave the response to even one State of the Union address? Outside of the person who gave it, the number who can answer "I remember" to that question is vanishingly small. If this were a movie, the President would be the star and the respondent would be — no, not a co-star, or supporting actor, but an extra. And who remembers extras?

Robusto's answer is perfectly right. I just wanted to add that while "ticket to oblivion" itself is not a very common phrase, the basic form "ticket to X" is a fairly well-established construction to refer to something that is certain to lead one to X. For example, "ticket to fame", "ticket to hell", "ticket to nowhere" and so on. I've heard a friend refer to an exceedingly greasy burger as a "ticket to cholesterol city".