From what I've read, the vast majority of Egyptian tombs were raided by grave robbers long ago, which is why the discovery of Tutankhamon's grave was so extraordinary, as it still contained the original treasures. But who actually did all the grave robbing? When did the practice begin, and what happened to the stolen treasure?
As you noted, Tutankhamun's tomb is virtually unique among the tombs of the kings for being mostly intact, though other tombs in the Valley of the Kings yielded important artifacts, most notably the tomb of Yuya and Tjuyu (KV 46) and the tomb of Maiherpri (KV 36). Most elite tombs were also robbed in antiquity; the far less famous tomb of Kha is one of the few tombs of the elite to escape unscathed.
Most textual accounts of tomb-robbing date to the 19th Dynasty and later. By the 19th Dynasty, the Egyptians had switched from building pyramids to building rock-cut tombs. The men responsible for constructing the rock-cut tombs in the Valley of the Kings and the Valley of the Queens lived in the workmen's village of Deir el-Medina, a walled settlement that was monitored carefully. Deir el-Medina was a state-supported village; the workmen and their families received a comfortable amount of food rations (barley, wheat, fish, honey, oil, etc.) and medical care in exchange for their work. It was to the king's benefit to pay his workers well; the workmen were prone to complain or even go on strike if their pay was withheld, and they were some of the few Egyptians to know the locations of the royal tombs.
Papyrus Salt 124, held in the British Museum, outlines the complaints of Amennakht about Paneb, the chief of workmen in Deir el-Medina. Amennakht accused Paneb of a litany of crimes, including attempted murder and rape. Among the charges was theft from a royal tomb:
Paneb's theft of King Seti Merneptah (Seti I). The list of them: ...and he took away the gilding of his chariot... of the doors. And they found four, but he took away one. It is in his possession...
And he stole the incense of the ennead of gods of the Necropolis and he divided it between himself and his fellows...
of inb-oil of Pharaoh and he took away his wines and sat on the sarcophagus of Pharaoh, although he was buried.
...one statue of the lord having upon it the name of Seti Merneptah and they took themselves off, and they were seen ...in the temple of Hathor, and the scribe Qenherkhepeshef confirmed what he had committed in the temple of Ptah, and Paneb...and chief workman Neferhotep. And he hacked up the ground which is sealed in the Place which is hidden.
And yet he took the oath, saying, "I did not upset a stone in the neighborhood of the Place of Pharaoh," so said he.
Paneb's ultimate fate is unknown, but he disappears from history after this, so we can assume he was punished harshly if not executed.
Given the unstable economic situation in the late 12th century BCE, people in the Theban area increasingly turned to tomb-robbing. Stolen goods were placed in one's own tomb or sold for profit, and rampant corruption meant that guards and officials could be easily paid off with bribes. Tomb-robbing became such a threat to royal tombs by the 21st Dynasty that priests secretly moved dozens of bodies from their tombs to DB320, where a cache of about 40 royal mummies was found in 1881.
The best information about tomb-robbing at the end of the New Kingdom comes from the "tomb robbery papyri." These are a dozen hieratic documents from the reign of Ramesses IX to Ramesses XI that describe the inspection of tombs and the proceedings of court cases involving accused thieves. The thieves had primarily focused on the 17th Dynasty tombs of Dra Abu el-Naga and the Valley of the Queens. Unsurprisingly, many of the people accused of tomb-robbing were inhabitants of Deir el-Medina.
The inspection found that some tombs had been undisturbed. From the Abbott papyrus:
The tomb of King Nebkheperre-Intef, L.P.H., it was found in course of being tunneled into by the thieves; they had tunneled 2½ cubits into its masonry, 1 cubit (distant) from the outer chamber of the tomb of the chief of the oblation-bearers of the House of Amun, Yuroi, which is in ruins. It was uninjured; the thieves had not been able to enter it.
Other tombs, however, had not fared as well.
The tomb of King Sekhemre-Shedtawy Sobekemsaf. It was found that the thieves had broken into it by mining through the base of it from the outer chamber of the tomb of the overseer of the granary of King Menkheperre (Thutmose III). The burial-place of the king was found void of its lord, as well as the burial-place of the great king's wife, Nebkhas, his royal wife, the thieves having laid their hand upon them. The vizier, the nobles, and the inspectors made an examination of it, and the manner in which the thieves had laid their hands upon this king and his royal wife was ascertained.
Accused thieves were beaten until they confessed to their crimes.
Then the vizier and the butler had the coppersmith taken before them to the tomb, while he was blindfolded. He was permitted to see (again) when he had reached them. The officials said to him: "Go before us to the tomb from which you said: 'I carried away the things.'" The coppersmith went before the nobles to one of the tombs of the king's-children of King Usermare-Setepenre (Ramesses II) in which no one was buried, which was left open, and (to) the hut of the workman of the necropolis, Amenemyenet, son of Huy, which was in this place, saying: "Behold, the tombs in which I was."
The nobles examined the coppersmith with a severe examination (i.e. beating) in the great valley, (but) he was not found to know any place there except the two places upon which he had laid his hand. He took an oath of the king that he should be mutilated (by cutting off) his nose and his ears and placed upon the rack (if he lied), saying: "I know not any place here among these tombs, except this tomb which is open, together with the hut upon which I have laid your hands."
The candor of the thieves is remarkable, and many confessed to disrespecting the remains of deceased rulers. An excerpt from the Amherst papyrus:
Nebkhas, his royal wife in the place of his... it being protected... with mortar, covered with blocks. We penetrated them, (and) we found her resting. We opened their coffins and their coverings in which they were. We found this august mummy of this king. There was a numerous list of amulets and ornaments of gold at its throat; its head had a mask of gold upon it; the august mummy of this king was overlaid with gold throughout. Its coverings were wrought with gold and silver, within and without, inlaid with every splendid costly stone. We stripped off the gold which we found on the august mummy of this god, and its amulets and ornaments which were at its throat, and the coverings wherein it rested. We found the king's wife likewise; we stripped off all that we found on her likewise. We set fire to their coverings. We stole their furniture, which we found with them, being vases of gold, silver, and bronze. We divided the gold which we found on these two gods, on their mummies, and the amulets, ornaments and coverings, into eight parts.
Tomb-robbing even appeared in Egyptian literature. In the Demotic tale of Setne, the eponymous character found out about a magical book buried in the tomb of Nanferkaptah, a long dead prince buried in the necropolis of Memphis. Setne, throwing caution to the winds, broke into the tomb and laid hands upon the book, whereupon the ghosts of Naneferkaptah and Ahwere materialized to warn him. The book, the couple informed him, had caused their deaths, as well as the death of their son. Undeterred, Setne seized the book. Upon reading the scroll, Setne encountered a beautiful woman named Tabubu. Tabubu convinced the smitten Setne to murder his sons and daughters, throwing their bodies to the dogs and cats. When Tabubu finally consented to sex with Setne, Setne awoke naked in front of the king, wholly embarrassed. Fortunately for Setne, the entire affair was merely an illusion, and the humiliated prince returned the book of magic to the tomb of Naneferkaptah before it could cause him further anguish.
Sources and further reading
"Papyrus Salt 124 (Brit. Mus. 10055)" by Jaroslav Černý, JEA Vol. 15, No. 3/4 (Nov., 1929), pp. 243-258
The Great Tomb-robberies of the Twentieth Egyptian Dynasty by T.E. Peet
Affairs and Scandals in Ancient Egypt by Pascal Vernus