By the time the Burr-Hamilton duel occurred, Jefferson and Burr were barely speaking even though Burr was the Vice-President. This is due to the fact that during the Election of 1800, there was a serious movement to place Burr as the President over Jefferson when the election went to the House of Representatives. Burr alienated Republicans by taking the position that he would not defer to Jefferson, the party leader, if he was selected as President.

Part of the hostility that ultimately led up to the duel between Hamilton and Burr was in fact b/c Hamilton actively endorsed Jefferson, whom Hamilton thought was mistaken in many areas, he at least had principles that he adhered to, over Burr, whom Hamilton viewed as having no principle but personal ambition, and his fellow Federalist Adams. Hamilton's endorsement was particularly powerful b/c even though he was on the downslope of his influence and power, he still controlled enough of the Federalist party to have the potential to swing a vote in the House and some have argued that Hamilton's influence was what swung the eventual deciding vote, Federalist James Bayard of Delaware, to pick Jefferson over Burr.

Jefferson never forgave Burr after this election and basically cast Burr out of the Democratic-Republican party. Jefferson so distrusted Burr that he shut Burr out of the administration, only meeting Burr for dinner once every two weeks and only allowing Burr to meet with his Cabinet once a year. Burr further broke with Jefferson when he sided with the Federalists over the repeal of the Judiciary Act. Burr was also distrusted by the Federalists, whom he courted to try and get back to power, and after his fateful duel with Hamilton he lost all influence with the remnants of that party.

Source: Alexander Hamilton by Ron Chernow

Thomas Jefferson's relationship with Burr seems to have been friendly up until 1800 (I seem to recall TJ writing that he'd always known Burr was a bad egg, but contemporary evidence from the 1790s suggests otherwise). There's no evidence that it was ever a particularly close friendship though, and their most famous entanglement was mostly due to geographical vote wrangling. Jefferson and Burr were running mates in the election of 1800 back when the winner became president and the second place person became vice president. Jefferson wanted Burr primarily to try to gain support for the ticket in northern states. We've changed the system since then because of the potential for messy outcomes such as 1800: Jefferson and Burr won the votes of the Democratic-Republican electors, which meant they were tied. This forced the election into the House of Representatives which then went through a long series of inconclusive votes. Before the election, everyone understood that Jefferson was supposed to be the presidential nominee. However, while Burr didn't overtly campaign for the presidency, he let it be known that he would accept the position if offered and otherwise was a bit more passive than you might expect from someone who wanted to be VP in good faith. Jefferson at first seemed to trust Burr, who used to belong to the opposing Federalist party and plausibly could have defected and taken advantage of the Federalist-run House had he not been loyal. Ironically, Alexander Hamilton, political enemy of Jefferson and staunch Federalist, was a large influence in convincing certain Federalists states to abstain from voting for Burr, thus giving Jefferson the Presidency.

After the election, Burr's reputation within his party was pretty poor. He was seen as an opportunist. Jefferson no longer trusted Burr. Jefferson pointedly ignored Burr's advice and shut out Burr's associates from appointments. Rather than wielding influence in the executive branch as Burr might have hoped, he ended up spending most of his time serving in the role as President of the Senate.

After the duel with Hamilton, I can only find evidence of two letters from Jefferson to Burr, both of which are formal and perfunctory. (This is despite the Federalist assertions that TJ and Burr became chummier than ever after the duel, since Burr killed Jefferson's enemy.) Absence of evidence is not evidence of absence, but it fits in with the general picture of their relationship: formed by convenience and discarded when Jefferson found he no longer had trust in nor benefit from his association with Burr.

Later on of course, in 1806 Jefferson was instrumental in having Burr arrested and tried for treason (for his entirely different bizarre plot known as the Burr conspiracy -- too convoluted for me to get into now). Jefferson also refused to produce government documents requested by Burr's defense. The friendship was clearly long dissolved by this time.

I just wrapped up a class on landmark cases in U.S. History and the Aaron Burr treason trial was one of the cases we covered in the class. To learn about the trial, we read Fallen Founder: The Life of Aaron Burr by Nancy Isenberg. Here are a few passages that I think should answer your question:

In Washington, he [Burr] faced his colleagues with the poise he was known for, and on occasion dined peaceably with President Jefferson. With Burr's power in New York much diminished, and the way ahead for the Virginia dynasty of presidents looking clearer, Jefferson was extremely cordial to the man who slew his longtime arch nemesis Hamilton.

Pg. 272

Finally, amid all the uncertainty, we know that Burr appealed directly to Jefferson for an appointment of some kind (the evidence is in Jefferson's private papers). The president, however, showed no inclination to assist his former vice president.

Pg. 301

Jefferson had been slow to react, but now he abruptly called the cabinet together, on October 22, 1806, to discuss Burr. Though he was relying on hearsay, he mentioned that Burr had "opened himself confidentially to some persons," and revealed his "scheme of separating the Western from the Atlantic States, and erecting the former into an independent confederacy." He undercut his supposedly objective assessment by mentioning that Burr had aroused "suspicions, as every motion does of such a Catalinarian [sic] character."

Pg. 307

Jefferson betrayed his feelings in his last private interview with Burr, in 1806. At that time, he told Burr he had lost the favor of the people, to which Burr responded, discerningly, that this might be true only if one relied on the newspapers. But Jefferson did rely on the newspapers--that is, on certain Republican papers--when it suited him. [...] In accepting select newspaper stories about Burr, he was unwilling to admit that he might have confused the people's will with his own.

Pg. 329

Jefferson's undisguised animosity was directed at two targets: the "crooked gun" Aaron Burr and the legal con artist John Marshall.

Pg. 338

So basically, Jefferson was ambivalent at best towards Burr, but once his allies convinced him that Burr was trying to take over the Western U.S. territories and turn them against the U.S., Jefferson lost all interest in doing anything other than openly hating Burr and doing everything he could to see Burr first locked up in prison, then second kept out of American embassies when Burr traveled abroad after the treason trial.