After Voldemort was resurrected, he was apparently immune to Harry's mother's protection thing. As I understand it, the reason Harry was with Dursleys was because he needed to be there to keep the protection. Did that protection break after GoF, and if so, why was Harry kept there? If not, why not?
Voldemort had the protection conferred by Lily’s sacrifice, but this was not the only protection on Privet Drive.
Details are sketchy, but Dumbledore extended Lily’s sacrificial protection when he left Harry with the Dursleys. Lily’s sacrifice alone does not protect Harry there, at least no more than any other location. As Dumbledore himself explains:
“The magic I evoked fifteen years ago means that Harry has powerful protection while he can still call this house home. However miserable he has been here, however unwelcome, however badly treated, you have at least, grudgingly, allowed him houseroom. This magic will cease to operate the moment that Harry turns seventeen; in other words, the moment he becomes a man. I ask only this: that you allow Harry to return, once more, to this house, before his seventeenth birthday, which will ensure that the protection continues until that time.”
— Half-Blood Prince, chapter 3 (Will and Won’t)
Voldemort only gets past Lily’s sacrificial protection. This does not allow him to bypass Dumbledore’s protective enchantments.
There are also several pragmatic reasons:
Voldemort is unlikely to attempt an attack on Privet Drive unless he can guarantee success.
At the end of the Triwizard Tournament, the Ministry are denying Voldemort’s return. Voldemort would like to maintain this illusion for as long as possible. If he attacks Harry, and fails, then he risks alerting the Ministry to his return.
Privet Drive is still fully protected against everybody except Voldemort.
It’s implied in Deathly Hallows that Dumbledore’s protection keeps out all manner of dark creatures, not just Voldemort:
“I hope so,” said Harry, “because once I’m seventeen, all of them — Death Eaters, dementors, maybe even Inferi — which means dead bodies enchanted by a Dark wizard — will be able to find you and will certainly attack you. And if you remember the last time you tried to outrun wizards, I think you’ll agree you need help.”
— Deathly Hallows, chapter 3 (The Dursleys Departing)
So Voldemort can’t dispatch the Death Eaters to kidnap or attack Harry, because they can’t attack Privet Drive. And if Voldemort were to stage an attack, he would be doing so alone.
Moving Harry leaves the Dursleys as vulnerable targets.
Were Harry to permanently move away from Privet Drive, he would no longer call it home. This means the Dursleys are no longer protected. I forget the reference, but I believe at one point somebody muses the possibility of Voldemort kidnapping and torturing the Dursleys to get to Harry.
If Dumbledore believes this is a legitimate possibility, then moving Harry would be very foolish.
Dumbledore has arrangements to protect Harry at Privet Drive.
As we learn in Order of the Phoenix, Dumbledore has people keeping an eye on Harry in Little Whinging. We know that Mrs. Figg has been doing so for years, and it’s possible that people have been monitoring Privet Drive since Harry was left there. Moving Harry means making alternative arrangements for his new destination. Dumbledore already has enough on his plate, so why change what works?