The Bible does not mention Nimrod's death. However in the Book of Jasher we read that he was killed by Esau, the son of Issac. Esau takes the coat of skins (Adam's garment) from Nimrod's body and flees. He is pursued by Nimrod's companions fearing for his life. At the end of his desperate escape, Esau goes to the home of his brother Jacob. Famished and thinking he is going to be killed at any moment, he barters the garment along with his birthright to his brother for a life-saving meal.
After Esau eludes the friends of Nimrod, they return and take his body back to Ur where he is probably buried in a regal public ceremony. Jasher notes that Nimrod was 215 years old at the time of his death. He had reigned for at total of 185 years.
Reading from the account of Jasher:
And Esau at that time, after the death of Abraham, frequently went in the field to hunt. And Nimrod king of Babel, the same was Amraphel, also frequently went with his mighty men to hunt in the field, and to walk about with his men in the cool of the day. And Nimrod was observing Esau all the days, for a jealousy was formed in the heart of Nimrod against Esau.....
And on a certain day Esau went in the field to hunt, and he found Nimrod walking in the wilderness with his two men. And all his mighty men and his people were with him in the wilderness, but they removed at a distance from him, and they went from him in different directions to hunt, and Esau concealed himself from Nimrod, and he lurked for him in the wilderness.....
And Nimrod and two of his men that were with him came to the place where (he was), when Esau started suddenly from his lurking place, and drew his sword, and hastened and ran to Nimrod and cut off his head. And Esau fought a desperate fight with the two men that were with Nimrod, and when they called out to him, Esau turned to them and smote them to death with his sword. And all the mighty men of Nimrod, who had left him to go to the wilderness, heard the cry at a distance, and they knew the voices of those two men, and they ran to know the cause of it, when they found their king and the two men that were with him lying dead in the wilderness.
And when Esau saw the mighty men of Nimrod coming at a distance, he fled, and thereby escaped; and Esau took the valuable garments of Nimrod, which Nimrod's father had bequeathed to Nimrod, and with which Nimrod prevailed over the whole land, and he ran and concealed them in his house.
And Esau took those garments and ran into the city on account of Nimrod's men, and he came unto his father's house wearied and exhausted from fight, and he was ready to die through grief when he approached his brother Jacob and sat before him. And he said unto his brother Jacob, Behold I shall die this day, and wherefore then do I want the birthright? And Jacob acted wisely with Esau in this matter, and Esau sold his birthright to Jacob, for it was so brought about by the Lord......
And when Nimrod the son of Cush died, his men lifted him up and brought him in consternation, and buried him in his city, and all the days that Nimrod lived were two hundred and fifteen years and he died. And the days that Nimrod reigned upon the people of the land were one hundred and eighty-five years; and Nimrod died by the sword of Esau in shame and contempt, and the seed of Abraham caused his death as he (Nimrod) had seen in his dream. And at the death of Nimrod his kingdom became divided into many divisions, and all those parts that Nimrod reigned over were restored to the respective kings of the land, who recovered them after the death of Nimrod, and all the people of the house of Nimrod were for a long time enslaved to all the other kings of the land. (Book of Jasher, Chapter 27)
There are other differing accounts of the death of Nimrod, however, after studying them all, I feel that this account of Jasher is the most reasonable and authentic.
It is very possible that Nimrod was buried in state with a great public ceremony and display of pageantry. It is also possible that he was accompanied in burial by a number of his family, servants, and staff. Such a burial was found by Leonard Wooley during his excavations at the site of Ur (Wooley, Excavations at Ur). He recorded, in great detail, all the different individual skeletons found there as well as all the rich treasures that were entombed with them. He originally thought that all these individuals had voluntarily sacrificed their lives to be buried with the royal personage, however, recent studies have found that they were murdered prior to burial and placed in order within the tomb. Whether the royal personage in this tomb could have been Nimrod is purely speculative, however, such a burial would have been consistent with his career.