The first thing to bear in mind is that both books - Isaiah and Revelation - are books of prophecy, and thus contain poetic and symbolic language. This means their interpretations should be considered carefully as much of the language is figurative. Moreover, the purpose of prophetic books are not simply to predict the future, but to express the redemptive plan of God in human history.
The two verses in question are (1) Isaiah 65:20...
No more shall there be in it, an infant who lives but a few days, or an old man who does not fill out his days, for the young man shall die a hundred years old, and the sinner a hundred years old shall be accursed.
And (2) Revelation 21:4...
He will wipe away every tear from their eyes, and death shall be no more, neither shall there be mourning, nor crying, nor pain anymore, for the former things have passed away.
For years, scholars have debated what Isaiah meant: (1) An intermediate "millennial" state or (2) the eternal state itself. The expression, "new heavens and a new earth" would seem to suggest an eternal state (such as the one described in Revelation 21:4), yet the mention of people dying, as well as the presence of sin, seem to suggest the opposite.
So what are we to make of this verse? The Bible Knowledge Commentary has a solid explanation:
In Revelation, however, the new heavens and new earth (Rev. 21:1) follow the Millennium (Rev. 20:4). Most likely Isaiah did not distinguish between these two aspects of God’s rule; he saw them together as one. After all, the Millennium, though 1,000 years in duration, will be a mere pinpoint of time compared with the eternal state.
Summarily, there is a period of time (the "millennium")* - described in Isaiah 65:20 - before the creation of new heavens and new earth - described in Revelation 21 by John.
Hope this wasn't (too) confusing. Remember, interpreting verses like Isaiah 65:20 are of importance, but not primary importance. What is of paramount importance is accepting Christ's gift of salvation while you still have time on this earth - before all of the events predicted by John & Isaiah occur.
[Answered by Pastor HM]
*According to this interpretation, one must concede that the millennial state includes both death and unbelief.