In the lawsuit, Mr Burton claims Carrey used his "immense wealth and celebrity status" to illegally obtain Ambien, Propranolol and Percocet that caused Ms White's death.
The complaint alleges that Carrey provided drugs to Ms White despite knowing she was "ill-equipped" to take them, prone to depression and she had previously tried to take her own life.
According to the amended lawsuit, filed by Mr Burton's lawyer Michael Avenatti, Ms White sent text messages to Carrey in 2013 saying that she suspected she had contracted an STI.
Carrey, 54, attempted to downplay her symptoms and later suggested she could have contracted them from someone else, the complaint states.
Mr Avenatti also shared a letter that White wrote Carrey in 2013 and says she has been called a "whore," an "opportunist," and an "ungrateful b****" by Carrey.
Burton released a statement saying he would not continue to "stand idly by while Mr Carrey and his handlers deceive the world about what they did to Cat."
"How dare they invoke her memory in an effort to save Mr Carrey's false persona," he said. "This is a search for the truth and people will now learn about the real Jim Carrey and his treatment of women.
"I urge other women who have been exploited and injured by him and his reckless behaviour to come forward and tell their stories," he said.
Where to get help
• Lifeline: 0800 543 354 (available 24/7)
• Suicide Crisis Helpline: 0508 828 865 (0508 TAUTOKO) (available 24/7)
• Youthline: 0800 376 633
• Kidsline: 0800 543 754 (available 24/7)
• Whatsup: 0800 942 8787 (1pm to 11pm)
• Depression helpline: 0800 111 757 (available 24/7)
If it is an emergency and you feel like you or someone else is at risk, call 111.