The Bible says that Jesus (as) ascended to heaven. Why do Muslims not believe this?

Because the account of the ascension is regarded even by Christian scholars as a later addition, and because it contradicts the promise of Jesus (as) to seek out the lost sheep of Israel.

Muslims believe in Jesus (as) as a truthful prophet. Precisely for that reason they look carefully at what later reports attribute to him.

The best known account of the ascension stands at the end of the Gospel of Mark. Christian biblical scholars largely agree that this ending was not part of the original text but was added later. The oldest versions do not tell of anyone seeing Jesus (as) rise up to heaven.

There is also a contradiction. Jesus (as) had promised that he would go to the lost sheep of the house of Israel. This meant the tribes of Israel who lived scattered far away. Had he ascended to heaven instead, he would have forgotten his own promise. One cannot assume that of a truthful prophet. When two statements contradict each other, one of them has to be given up.

Moreover, the Gospels themselves report that after the crucifixion the disciples saw Jesus (as) on earth, spoke with him and ate with him. This fits a man who came down from the cross alive, not someone who had left his body behind.

The Holy Qur’an says that God exalted Jesus (as) to Himself. This is an exaltation in rank, as is granted to everyone close to God. The Messengers were human beings who ate and drank and did not live forever. The Holy Prophet Muhammad (saw), too, when asked to ascend into heaven, answered that he was only a man and a Messenger.