What was the downfall in Tamburlaine the Great?

Christopher Marlowe’s tragedy Tamburlaine the Great was written in two parts around 1587 or 1588. The play is based on the life of Timur, also known as Amir Timur and Tamerlane, the founder of the Timurid Empire, in and around what is now Iran and Central Asia.
Once a shepherd and bandit, Tamburlaine rose far above his lowly state in life to become the Emperor of Persia, the Emperor or Turkey, and the Emperor of Africa. Tamburlaine was called "The Scourge of God' for his ruthlessness and brutality. By the time of his death, Tamburlaine was well on his way to becoming the emperor of the world.
Certainly Tamburlaine was ambitious, and his greatest flaw—like that of many Greek and Elizabethan tragic heroes—was his excessive pride.
Tamburlaine claimed that the gods blessed his conquests:

TAMBURLAINE: And so mistake you not a whit, my lord;
For fates and oracles of heaven have sworn
To royalize the deeds of Tamburlaine,
And make them blest that share in his attempts . . .

He was convinced of his own greatness . . .

TAMBURLAINE. Though Mars himself, the angry god of arms,
And all the earthly potentates conspire
To dispossess me of this diadem,
Yet will I wear it in despite of them,
As great commander of this eastern world,
If you but say that Tamburlaine shall reign . . . .
So; now it is more surer on my head
Than if the gods had held a parliament,
And all pronounc'd me king of Persia. . . .
I that am term'd the scourge and wrath of God,
The only fear and terror of the world . . .

Tamburlaine burned the holy book of Islam, the Qur'an . . .

TAMBURLAINE. Now, Casane, where's the Turkish Alcoran,
And all the heaps of superstitious books
Found in the temples of that Mahomet
Whom I have thought a god? they shall be burnt . . . .

And he declared himself to be greater than the gods:

TAMBURLAINE: The god of war resigns his room to me,
Meaning to make me general of the world:
Jove, viewing me in arms, looks pale and wan,
Fearing my power should pull him from his throne:
Where'er I come the Fatal Sisters sweat,
And grisly Death, by running to and fro,
To do their ceaseless homage to my sword . . .

Tamburlaine even killed his own son, Calyphus, because he didn't share Tamburlaine's ruthless, unforgiving attitude towards war and the people he conquered.

TAMBURLAINE. Stand up, my boys, and I will teach ye arms,
And what the jealousy of wars must do . . . .
Here, Jove, receive his fainting soul again;
A form not meet to give that subject essence
Whose matter is the flesh of Tamburlaine,
Wherein an incorporeal spirit moves,
Made of the mould whereof thyself consists,
Which makes me valiant, proud, ambitious,
Ready to levy power against thy throne,
That I might move the turning spheres of heaven;
For earth and all this airy region
Cannot contain the state of Tamburlaine.
[Stabs CALYPHAS.]
By Mahomet, thy mighty friend, I swear,
In sending to my issue such a soul,
Created of the massy dregs of earth,
The scum and tartar of the elements,
Wherein was neither courage, strength, or wit,
But folly, sloth, and damned idleness,
Thou hast procur'd a greater enemy . . .

Unlike the tragic flaws of other Greek and Elizabethan tragic heroes, however, Tamburlaine's pride and ambition (and all of his other many character flaws) didn't prove to be his downfall, and least not in this world. Tamburlaine was never defeated in battle. Only illness and death conquered him.
As to where Tamburlaine might have gone after his death, we can only surmise from the extent of his pride, deceit, vengefulness, violence, savagery, and tyranny while he was alive.