Hello, What happens if the ground wire is not connected to the ceiling light fixture?

For most light fixtures the lights will work OK even if the ground is not connected. However, the ground is provided as a safety to ensure that the fixture chassis is grounded - which can prevent possible shock if there is ever a short between an energized wire and the body of the fixture.

If there is not a separate ground wire or grounding attachment point on the fixture, then in effect you are relying on the mecahnical attachment of the fixture via its mounting screws to the crossbar as the ground path. So make sure that the mounting box and the mounting crossbar both are securely connected to ground.