I wonder, how accurate the lunar calendar is? Can we predict the new moons precisely for tens of years in future? If we know all factors that affects the movement of the moon, then why it is not possible to predict it?
The motion of the moon is known with great accuracy, and so the exact moment of a new moon (the moment that the sun and moon have the same longitude in the sky) can be calculated with great accuracy for tens of thousands of years into the future. This is the astronomical meaning of New moon, and the only one which is relevant here. The uncertainty is due to effects such as the non-spherical nature of the Earth's gravity and the exact magnitude of perturbations of the orbit by other bodies.
Unless there is an eclipse, the moon is not visible at the moment of the new moon, and doesn't become visible until some time later (How Soon Could a Waxing Crescent Moon Be Seen?). The time when the moon can be seen for the first time depends on lots of factors, such as atmospheric clarity, weather and latitude, these can't be predicted from day to day.
The answer depends on what you mean with "precisely". The ELP theory seems to have an accuracy of ~10-6° in longitude over a century, which comes down to about a millisecond for phases.