Treat this as a balanced budget question, so it's not about fiscal policy. Alternatively, imagine yourself as a benevolent philanthropist: should you support this area of research if you can do so as a free lunch? Or should you try to hinder it?
I believe no one understands the underlying science much at all. But there is some chance that the old science fiction movies are correct and that by time-traveling you alter the course of history, thereby obliterating the universe we used to have. I'll count that as a net negative, while noting there is some chance we end up with a better universe.
On the plus side, the human race will die out anyway. Time travel seems to yield a fairly safe haven. As disaster approaches, keep going back in time a few days, or decades, and that asteroid will never hit you. This is especially appealing if you are transporting back a body (upload?) which is programmed to be more or less immortal and you can take the technology with you, so as to keep on going back as time progresses.
On one side: immortal life for many of the last humans and thus immortality for the human race. And with time they may learn how to thwart the asteriod. On the other side: some probability of swapping universes.
So should we subsidize or tax research into time travel?