Unfortunately, the answer to this question is unfulfilling. Here's the story in relative fullness.

A French priest by the name of Peter Bartholemew, a known drunkard, claimed he had had a vision, given to him by St. Andrew, of Christ on the Cross being pierced with a lance that could be found within the city of Antioch. Despite initial skepticism among the Crusade leaders, Raymond of Toulouse, the overall commander, initiated a search of the Cathedral of St. Peter, and Peter Bartholemew eventually managed to produce an authentically antique Lance (the head of one at least) from a pit within the church. The Lance provided a source of inspiration for the flagging Crusaders, and Raymond, seeing the increasingly desperate situation and firmly believing in the provenance of the Lance, ordered the army to attack the enemies outside the walls of Antioch despite being outnumbered. Bishop Adhemar Le Puy, leading one of the flanks of the army into battle, actually carried the Lance with him as a standard (despite being skeptical about its authenticity). The Crusaders won the battle, due in large part to morale and command issues within the army of their enemy, and many attributed the victory to the holy blessing of Peter's Lance. Per at least one eyewitness account,

“The lines of the enemy fell upon us who were in the squadron of the Bishop, and though their forces were greater than ours, yet, through the protection of the Holy Lance which was there, they there wounded no one; neither did they hit any of us with arrows.”

There were also reports of holy spirit knights joining the Crusader forces on the charge and more besides. However, despite the excitement over the Lance, critics among the Crusader forces remained. The most outspoken was Bohemond of Taranto, a rival of Raymond of Toulouse, who publicly claimed that Peter was a fraud. Peter, indignant at these claims and fearing that he would lose the power and following he had gained as a result of the lance's discovery, insisted that he would undertake a literal trial by fire to prove his holiness and the authenticity of the Lance. He proclaimed in front of the assembled leaders of the Crusader force that:

“I wish and beg that a very large fire be built; and I will pass through the midst of it with the Lance of the Lord. If it is the Lance of the Lord, I will pass through the fire unhurt, but if it is not, I will be burned in the fire. For I see that neither signs nor witnesses are believed.”

With the approval of all present, an incredibly large fire was built, and Peter made to pass through it with Lance in hand. Remarkably, he did survive the flames despite being horrifically burned, though he died 12 days later in agony. Though Peter had survived his trial, his horrible burning was enough evidence that the Lance was not the divine protection it was made out to be.

The exact fate of the Lance is not known and is not really discussed in the sources describing the incident. This lack of sources on the subject is probably attributable to the fact that the Lance lost its holy glamor after the trial by fire (and no longer had any sway politically). As important as it was at the time, after the victory at Antioch it no longer held significance for the majority of the Crusader host. From what we know, it does not appear that it was destroyed in the blaze that claimed Peter, though it may well have been damaged beyond recognition. One Ferdinand de Mely in the early 20th century argued that spearhead (more a bannerhead) that he had found in Etchmiadzin Cathedral in Armenia was likely the Lance of Peter Bartholemew. Another theory is that the Lance was kept in Antioch and later carried by the Crusader host to Jerusalem. Finally, there are fanciful murmurings that the Lance was kept in Antioch and captured by the Turks, who somehow later returned it to the Vatican. Truly, the only confirmed thing we know is that Raymond of Toulouse was in possession of the Lance after the trial by fire and that it disappeared from sources thereafter. Other relics, such as a piece of the True Cross, soon take precedence in the Crusader host. This is speculation on my part, but I imagine that, because of the potential embarrassment it could cause Raymond if his rivals had probed at its legitimacy further and found it even more wanting, the Lance was bundled away into a storeroom somewhere or simply discarded quietly, perhaps even reburied in the cathedral at Antioch.

TLDR: Short answer: We don't know what happened to it really. Long answer: A number of things could've happened to it, but the most likely outcome is that it was discarded when it was decided it wasn't really that holy.

Sources:

The Holy Lance of Antioch: A Study on the Impact of a Perceived Relic during the First Crusade by Marius Kjørmo http://bora.uib.no/bitstream/handle/1956/3327/56487859.pdf;jsessionid=A3A354B0A5FB9A28EEAFB94199A1C118.bora-uib_worker?sequence=1

The Holy Lance of Antioch by Adam Nemeroff http://sites.dartmouth.edu/crusadememory/2016/04/24/the-holy-lance-of-antioch-2/

The Discovery of the Holy Lance by Richard Cavendish http://www.historytoday.com/richard-cavendish/discovery-holy-lance

The Holy Lance of Peter Bartholemew (a collection of sources pertaining to the incident on a strange website) http://home.ix.netcom.com/~kiyoweap/myth/arms-weap/holy-lance-peter-bartholomew.htm