No, Aaron Burr did not sign the Declaration of Independence. Although he was a prominent political figure in the early days of the United States, serving as vice president under Thomas Jefferson, he was not one of the signatories of this foundational document.
The Declaration of Independence was adopted on July 4, 1776, by the Continental Congress, with the document being signed by 56 delegates from the thirteen colonies. At that time, Burr was not yet involved in the national political scene; he was just a young man pursuing a legal career.
Burr did become significantly involved in politics later on, but his legacy is more often associated with his duel with Alexander Hamilton and his subsequent trial for treason, rather than the founding documents of the United States.