Summary: The Qur'an says Satan deceived Prophet Adam and his wife: what Satan says is unreliable. Unlike Satan, Allah just refers to it as "the tree". The Qur'an says people dwell in heaven forever, and everyone on Earth will perish (which includes Prophet Adam and his wife).
Satan was deceiving Prophet Adam and his wife (Qur'an 7:22 So he made them fall, through deception.), so it's likely that he lied or told a half-truth.
Moreover, Allah refers simply to a "tree" (شجرة):
And We said, "O Adam, dwell, you and your wife, in Paradise and eat therefrom in [ease and] abundance from wherever you will. But do not approach this tree, lest you be among the wrongdoers."
And "O Adam, dwell, you and your wife, in Paradise and eat from wherever you will but do not approach this tree, lest you be among the wrongdoers."
This is different to how Satan refers to it:
But Satan whispered to them to make apparent to them that which was concealed from them of their private parts. He said, "Your Lord did not forbid you this tree except that you become angels or become of the immortal."
Then Satan whispered to him; he said, "O Adam, shall I direct you to the tree of eternity and possession that will not deteriorate?"
Besides, Prophet Adam and his wife did not gain immortality from eating from the tree:
Everyone upon the earth will perish,
And the Qur'an says people in Heaven are there forever:
But those who believe and do righteous deeds - We will admit them to gardens beneath which rivers flow, wherein they abide forever. For them therein are purified spouses, and We will admit them to deepening shade.