Ignoring the concept of "can we apply that much delta-V to a planet?", I'd be interested to know whether it's believed that there exists a "Goldilocks zone" suitable for life at all stages of a star's life. Intuitively it seems like there should be, I'm not sure.
Of course, it should be pointed out that the common understanding of "when the sun becomes a red giant" may be a bit flawed; the sun will cool and expand, then collapse. On a human time scale, it will spend a lot of that time as a red giant, but if you simply took the Earth when its orbit started to be crowded by the inner edge of the Goldilocks zone and put it in a new orbit, that new orbit wouldn't be anywhere close to an eternally safe one. Indeed, I suspect that the outermost of the orbits required for the giant-stage sun would be too far from the sun at the time we'd first need to move the Earth.
This thread is for asking any questions that might seem obvious, tangential, silly or what-have-you. Don't be shy, everyone has holes in their knowledge, though the fewer and the smaller we can make them, the better.
Please be respectful of other people's admitting ignorance and don't mock them for it, as they're doing a noble thing.
To any future monthly posters of SQ threads, please remember to add the "stupid_questions" tag.