In "Harry Potter and the Prisoner of Azkaban" Harry and Hermione saved two lives (Sirius and Buckbeak). And in "The Order of Phoenix" Sirius died by Harry's fault. Why didn't Harry use The Time-Turner in that time to take him back again?
There are a couple of reasons:
He wasn't thinking
His fist action after Sirius's death was to try to kill Bellatrix; he wasn't logically thinking at that moment. He wouldn't have thought about using Time-Turners.
Sirius would resist.
If Harry appeared randomly and tried to save Sirius, he would be suspicious of some kind of Dark Magic. Or, if he didn't get suspicious, he would turn the tables and try to protect Harry, at which point nothing would be different.
He didn't have a Time-Turner
Hermione gave hers back at the end of 3. Yes, there were Time Turners in the Department of Mysteries, but:
...while the glass cabinet that Harry now suspected had contained Time-Turners continued to fall...
“Ar, I always knew yeh’d find it hard ter squeeze me inter yer timetables,” he said gruffly, pouring them more tea. “Even if yeh applied fer Time-Turners —”
“We couldn’t have done,” said Hermione. “We smashed the entire stock of Ministry Time-Turners when we were there last summer. It was in the Daily Prophet.”
Even if he had thought of it, he couldn't have. Even if he could have, it's unlikely that it would have worked.
In PoA, it was possible to save him without altering established events.
Harry and Hermione hadn't seen Sirius die in HP and the Prisoner of Azkaban. They knew he was imprisoned, but he wasn't dead yet; the only reason they needed to go back in time in order to save him was so that they could set everything up to give themselves that possibility.
They hadn't seen Buckbeak die either. They thought they'd heard the executioner killing him, but as they discovered 'later', it was actually something else they'd heard. So even Buckbeak, whom they really did have to go back in time for, could be saved without changing anything they'd seen.
In OotP, saving Sirius would have caused a time paradox.
Sirius died right in front of Harry. The only way of saving him using a Time-Turner would have been to go back in time and interfere with established events. Harry would have had to jump out of somewhere to change what he already knew had happened, right under his slightly younger self's nose. There would have been no way of doing it without causing a time paradox. As Hermione says in PoA, time paradoxes are something to be avoided at all costs when using Time-Turners.
In the Harry Potter universe, time is a fixed entity - that is, you cannot create a paradox. There is only one timeline.
In Prisoner of Azkaban, Harry and Hermione were able to go back and save Sirius and Buckbeak because there was no paradox preventing them from doing so. Buckbeak never died, and Sirius was always going to escape - They didn't change the timeline because they'd already done it!
Remember once before Harry and Ron knew about the Time Turner, and Hermione was using it to get to class. She missed Charms one day, and the two of them found her sleeping in the common room. She couldn't just use the Time Turner to go back to Charms, because she'd already missed it!
The same goes with Sirius in Order of the Phoenix. Harry saw him die, so couldn't have gone back and prevented it. That would be a paradox.
And on top of that, let's not forget that the Ministry of Magic's entire supply of Time Turners was destroyed during the battle with the Death Eaters!