The mishna in Sanhedrin 90a lists Gehazi as one of the 4 commoners who have no share in the world to come. I have seen different reasons for this none of which I fully understand.
He mocked the words of the sages - artscroll says this, not sure when he did this but its not included as one of the reasons why someone would lose their olam haba in the mishna
he denied the resurrection - this is quite possible according to the midrashim that say he spoke (disparagingly) on his way to revive the shunamite woman's son. But later on he extols his master's ability to resurrect that same child to the king. So even if he may have doubted in the past he clearly no longer does.
He pronounced the name of God - this is my own interpretation which I haven't seen elsewhere. The mishna lists that as one of the reasons to forfeit olam haba [according to one opinion] and Gehazi seems to have done something very similar to make Yerovam's calf speak (see Sotah 47a)
He didn't repent for causing others to sin - found in otzar aggados - similarly not listed in the mishna.
I'm looking for a clear-cut reason why Gehazi has no share in the world to come.
the relevant discussion is in Sotah 47a (quoting from std english translation):
R. Johanan said: He (Elisha) went to induce Gehazi to repent but he refused. He said to him, ‘Repent’; but he replied: ‘Thus have I received from thee that whoever sinned and caused others to sin is deprived of the power of doing penitence’. What had he done?
Some say: He applied a loadstone to the idolatrous image of Jeroboam and suspended it between heaven and earth.
Others say: He engraved upon it the Name [of God] so that it used to exclaim, ‘I [am the Lord thy God]’ and ‘Thou shalt have no [other God beside me]’
Still others say: He drove the Rabbis from before him, as it is written: And the sons of the prophets said unto Elisha, Behold now, the place where we dwell before thee is too strait for us — hence, up to then it had not been too strait.
It seems like he was prvented from doing teshuva for some sort of chillul Hashem which does not allow him to get his share in the World to Come. I did see in lectures by Rav Moshe Wolfson that there are chassidic sources which disclose that at the end of days, Elisha will take back the curse and Gehazi will repent.