This doesn't mean that he can never lie. Buddhists who take the fourth precept are still humans. Could be that you fail to keep the precept - either deliberately or accidentally, because as humans we sometimes fail at things.
That's like stumbling as you walk.
Buddhists can also lie out of compassion to prevent some harm. For a clear example, you might tell a lie to save someone from murder, and to save the murderer also from his or her action.
However when we call someone a liar - normally that doesn't include lies said out of compassion. Rather it usually means someone who uses speech to deceive others with harmful intent towards them - or motivated by a wish to cover up their own inadequacies and failings.
Someone who wants to follow the Buddha's path does what they can to avoid conduct like that.
But - he's not a perfect person - whether you think of him as an ordinary person, like anyone else, following the path, or have a special connection and find him especially inspiring - just by being in this world with us, that makes him imperfect and facing many difficult and confusing and complex situations - with no perfect response possible to some of them.
What's more important, than checking to see if someone else is telling the truth, in everything they say (how can we ever know for sure if everything they say is truthful) is to refrain from harmful and deceitful speech oneself. And then to investigate the truth for oneself as best one can.
That's how the Buddha taught his followers, not to rely on any one else as an ultimate source of truth, on anything else, nor any book or inanimate thing, but to investigate and check things out and to find out for themselves. Others can only be an inspiration and starting point, to point you in the direction of discovery.
For some, the Dalai Lama is an inspiration and helps them to find the path to truth in themselves.
As for those who accuse him of lying on this and that - easy to find with a google search - well - it's interesting, it seems to be rather one way. I haven't heard of him accusing others of lying. And often the spotlight in something like that falls more on the accuser than the accused.