Did some quick research on this. First, this is the text of the relevant section of the Constitution in operation at the time. The 12th Amendment was passed afterwards to prevent this situation from happening again:
"The Electors shall meet in their respective States, and vote by Ballot for two Persons, of whom one at least shall not be an Inhabitant of the same State with themselves. And they shall make a List of all the Persons voted for, and of the Number of Votes for each; which List they shall sign and certify, and transmit sealed to the Seat of the Government of the United States, directed to the President of the Senate. The President of the Senate shall, in the Presence of the Senate and House of Representatives, open all the Certificates, and the Votes shall then be counted. The Person having the greatest Number of Votes shall be the President, if such Number be a Majority of the whole Number of Electors appointed; and if there be more than one who have such Majority, and have an equal Number of Votes, then the House of Representatives shall immediately chuse [
sic] by Ballot one of them for President; and if no Person have a Majority, then from the five highest on the List the said House shall in like Manner chuse [
sic] the President. But in chusing [
sic] the President, the Votes shall be taken by States, the Representation from each State having one Vote; A quorum for this Purpose shall consist of a Member or Members from two thirds of the States, and a Majority of all the States shall be necessary to a Choice. In every Case, after the Choice of the President, the Person having the greatest Number of Votes of the Electors shall be the Vice President. But if there should remain two or more who have equal Votes, the Senate shall chuse [
sic] from them by Ballot the Vice President."
The election was decided by the House of Representatives because of a coordination failure (or trick by the Burr faction) so that all the Jefferson electors voted for both Jefferson and Burr, meaning they both got 73 votes, while one of the Adams electors did not vote for his running mate, Pickney, so Adams came third with 73.
There apparently was confusion at the time as to who would become Vice President after the House of Representatives chose the President. Though the drafting of the 1787 Constitution was horrible, to me the text clearly reads that once the House chooses the President, the candidate with the most electoral votes who did not become President would be Vice President, which IOTL in fact was Burr, and if Burr had been elected would have been Jefferson. Jefferson already was Vice President, having gotten the office through a similar coordination failure on the part of the Federalists in the previous election. The procedure was really horrible until the 12th Amendment sort of fixed it.
One change the 12th Amendment made was to allow the House of Representatives to choose the third place finisher in the Electoral College, had this been operational in 1800 they could have picked Adams.
To complicate things further, there was the idea among the Federalists of just maintaining the deadlock in the House. The relevant Wikipedia article states that John Marshall, then Secretary of State, would have become acting president until the deadlock was resolved, which might have taken until the 1802 congressional election. However, the Presidential Succession Act of 1792 made the President Pro Tempore of the Senate next in line, and the Senate not having selected a President Pro Tempore, the next in line after that was Theodore Sedgwick, Speaker of the House. The Presidential Succession Act provided that whoever became acting president would serve as such until a new permanent president was chosen. Marshall would only be appointed and confirmed to the Supreme Court IOTL after the House elected Jefferson.
If the Federalists had just elected Burr, then constitutionally Burr becomes President, and Jefferson remains as Vice President, whether he takes the oath or not. If Jefferson resigns as Vice President, the next in line for presidential secession would have been whoever the House elected as Speaker (IOTL Nathanial Macon), though this could have been Jefferson himself, and the Senate could still have elected a President Pro Tempore.
Politically, Burr's election would have been corrupt bargain of 1824 on steroids (actually the 1824 election of Adams was completely above board and entirely in accordance with the rules, and unlike Jefferson, Jackson never had an Electoral College majority). However, IOTL, there have been two instances where a Vice President of a different party than the President became President due to a vacancy (1841 and 1865), two instances where a President was assassinated and there were suspicions that that his successor or his followers were behind the assassination (1881 and 1963), and four or five instances of an Electoral College/ Popular Vote reversal (1876, 1888, 2000, and 2016, possibly also 1960), with one of them (1876) being highly suspicious and needing to be resolved by another bargain between the parties. Based on these examples, Burr would have become President, though probably serving only one term. There might be a genuine three way contest for the Presidency in 1804, but by then the 12th Amendment or something similar would have passed.
Where this becomes interesting is how Burr handles negotiations with Napoleon and the Louisiana purchase.