I don't think Harry would automatically consider Dementors jumping into the Sun an optimal thing to do as there are too many unknowns.

• Can a Dementor fly high enough or fast enough to leave the Earth's atmosphere? Practically speaking, we don't know how much of their flight is true flight and how much they are bound by forces like gravity (not as much as true material beings, obviously, but that's not necessarily the same as not at all).

• Can a Dementor survive away from the Earth? We don't know if they need to absorb sustenance. It may be that they need to draw on the Earth's ambient magical field to continue to exist, or to remain within a certain range of living beings with thoughts and feelings to passively absorb. It may be that shadows of Death have no meaning too far away from the nearest living thing.

• How long would it take a Dementor to reach the Sun? We don't know their maximum travelling speed, but you can Apparate away from them, and the journey might easily take long enough for humanity to achieve magico-technological singularity in the meantime, or for cosmic-scale Patronus casting to become a thing, or any number of other solutions to be implemented.

• Can Dementors destroy the Sun? Azkaban is still standing despite having a significant concentration of Dementors in it for at least decades, and we know it's not because it's magically protected, since Dementors consume magic. It's plausible that all the Dementors on Earth will not have a sufficient impact on the Sun's lifespan to make a difference to humanity.