We know that Earth That Was became difficult to live on, so humanity spread out to terraform and colonize other worlds. Was Earth TOTALLY abandoned, or is it just a difficult environment?
There are a couple of canon descriptions of the ultimate fate of "Earth-That-Was", neither of them it must be admitted, from especially reliable sources
The implication is clearly that the Earth's atmosphere became uninhabitable and may have been on the verge of being lost entirely.
The second canon reference is seen in the film "Serenity" and is from an even more dubious source, an Alliance teacher;
While she is speaking, we see the image above. The Earth appears to be in very bad shape. The atmosphere is visibly brown, the seas are almost black and the landmasses no longer seem to be capable of supporting vegetation. There is visible sea-level rise and there are columns of reddish clouds.
Although I'd bow to the judgement of an actual climate scientist (and taking into account that the Alliance teacher might simply be lying or the graphic faked), I find it hard to believe that an atmosphere with that much contamination could actually sustain human life.
I don't recall this being settled on-screen one way or the other.
On the one hand, it would be odd if Earth was completely abandoned because of environmental degradation. It is humanity's original home, after all. Even if people needed artificial environments to survive, you would expect a few people to hold on there. And if they can terraform other worlds, surely they could use the same technology to fix Earth's environment, even if it took a long time.
Then again, there is absolutely no mention of ongoing contact with Earth. If anyone was still there, you would expect them to be able to send radio signals back and forth. And they always refer to "Earth-That-Was", not "Earth-That-Is-Far-Away".
It could be that the school lesson we see at the beginning of the film Serenity was just sanitised Alliance propaganda, and the real reason for leaving Earth was more alarming. Perhaps the planet was made uninhabitable by out-of-control nanotechnology, a robot uprising, or the like. Or maybe the settlers in the Firefly universe were deliberately exiled as criminals, religious or political dissidents.
We can speculate but I don't think we really know.
What little knowledge of Earth That Was seems to be summed up here.
The important part being:
There might be some apocryphal stuff from other sources like the RPG game.
The only other relevant information I could recall from the show not already posted here pertains to the Lassiter, one of the first laser pistols built on Earth-That-Was. Saffron and Serenity's crew set out to steal it in the episode "Trash" (more info about the Lassiter and this episode here), and its previous owner is a renowned collector of ETW artifacts. There's some implication in this episode that artifacts like the Lassiter were passed down through generations and not recently recovered from ETW. However, this is just more speculation. To my memory, the crew encountered no empirical evidence of the fate of ETW.