JK Rowling has explained that Veritaserum is not routinely used in court cases because it is "an unfair and unreliable tool to use at a trial" (bold text mine):
Q. Veritaserum plays a big part in finding out the truth from Mad-Eye Moody in book four. Why then is it not used for example in the
trials mentioned in the same book? It would be much easier in solving
problems like whether Sirius Black was guilty or not?
JKR : Veritaserum works best upon the unsuspecting, the vulnerable and those insufficiently skilled (in one way or another) to
protect themselves against it. Barty Crouch had been attacked before
the potion was given to him and was still very groggy, otherwise he
could have employed a range of measures against the Potion - he might
have sealed his own throat and faked a declaration of innocence,
transformed the Potion into something else before it touched his lips,
or employed Occlumency against its effects. In other words, just like
every other kind of magic within the books, Veritaserum is not
infallible. As some wizards can prevent themselves being affected, and
others cannot, it is an unfair and unreliable tool to use at a trial.
Sirius might have volunteered to take the potion had he been given the chance, but he was never offered it. Mr. Crouch senior, power mad
and increasingly unjust in the way he was treating suspects, threw him
into Azkaban on the (admittedly rather convincing) testimony of many
eyewitnesses. The sad fact is that even if Sirius had told the truth
under the influence of the Potion, Mr. Crouch could still have
insisted that he was using trickery to render himself immune to it.
While Harry is far less skilled of a Wizard than Barty Crouch Jr. or Sirius Black, it stands to reason that, because there is the possibility of fakery (especially with someone as powerful as Dumbledore on his side), the Wizengamot on principle does not administer Veritaserum to defendants.
And even if the Wizengamot could have considered Veritaserum for Harry, consider too that the Ministry does not want to encourage Harry's story that Voldemort is back. If he's given Veritaserum and starts repeating his story, that makes it harder for the Ministry to deny.