Edward Gibbon seems to be the source of the myth about "Christians destroying the Great Library", via his version of the story of the dismantling of the Serapeum, which had (at one stage) held a smaller "daughter library" that survived the end of the actual Great Library by around a century:

"The valuable library of Alexandria was pillaged or destroyed; and near twenty years afterwards, the appearance of the empty shelves excited the regret and indignation of every spectator whose mind was not totally darkened by religious prejudice. The compositions of ancient genius, so many of which have irretrievably perished, might surely have been excepted from the wreck of idolatry, for the amusement and instruction of succeeding ages.” (Gibbon, The History of the Decline and Fall of the Roman Empire, Vol. V, Ch. 28)

The only problem here is that it seems the Serapeum no longer contained any library when it was destroyed on the order of the emperor Theodosius after it had been used as a bastion by a group of radical pagans who terrorised and murdered Christians. So Gibbon's claim about indignant spectators looking on its empty shelves is pure fantasy. Unfortunately, his story appealed to many peoples' prejudices and it was given special prominence by Carl Sagan in his popular 1980 TV series Cosmos, and so it has become fixed in popular consciousness despite being almost entirely nonsense.

See History for Atheists - The Great Myths 5: The Destruction of the Great Library of Alexandria for details of how the myth arose and why it is wrong.