As bizarre as it sounds, Burr's relationship with them actually improved afterwards.

This is actually a good followup question to another answer that I posted on the AMA thread today about the 1804 gubernatorial campaign.

So following that defeat, the duel, and the threat of indictment in New Jersey, Burr went off into the South and was essentially a fugitive until November, although one interesting aspect to that initial exile was that when he periodically popped his head up he was often feted for killing Hamilton; in the border regions, the latter's death was celebrated rather than mourned as it was elsewhere, and that outpouring of support among the region's largely Jeffersonian Republican voters may have been noted by their almost entirely Jeffersonian Republican elected officials.

But there was something else going on too. In early 1803, the House voted for the first time to impeach a federal judge, District Court Judge John Pickering, who had been appointed by Washington in 1795 but had deteriorated so noticeably after 1800 that he was considered insane and a temporary replacement had been named. While pretty much all parties acknowledged that he needed to be removed from the bench, the controversy over his impeachment in the House in March 1803 and Senate conviction in early March 1804 - the delay was one of the downsides of 19th century Congresses holding their regular sessions only from November or December to March - was that Federalists vehemently insisted that impeachment was not an acceptable method to achieve it, correctly recognizing that Jefferson's next usage would be as a political weapon against some of the Federalist judges appointed by Adams.

Their prediction was accurate. A couple weeks later, in late March 1804 shortly before the first session expired, the House of Representatives voted to impeach Supreme Court Justice Samuel Chase. Given the Congressional schedule, though, no trial could take place until the next session in November, and the presiding officer of the trial would be...one Aaron Burr, still Vice President of the United States and President of the Senate.

Chase was a piece of work. A High Federalist, he'd run his courtroom viciously, bullying witnesses and attorneys he disagreed with and ruling arbitrarily, which comprised most of the charge sheet laid against him by the House. More significantly to Republicans (although not in the Articles of Impeachment presented), he was also someone who 5 years earlier while riding circuit had been the single most rabid of the Sedition Act judges, in the process jailing and attempting to deliberately bankrupt a good deal of the operators of the Republican press. Chase did not even remotely make an attempt towards either justice or impartiality in his rulings; one particularly egregious incident was when friends attempted to pay a publisher's fine, Chase escalated it by something like 10 times and threatened to do so again if the money didn't come directly out of the pocket of the publisher.

So Jefferson had more than a few longstanding grudges against Chase, and trying to enlist Burr to help at the trial was viewed as far more important than the fact he'd killed Hamilton. Burr had largely been on the outs with the administration for the better part of three years, but suddenly he was lavished with attention. Once he was back in November with Congress in session, Jefferson dined with him on multiple occasions, Madison took him along to a meeting with the French ambassador, the Senate Republicans drafted a resolution urging New Jersey's governor to quash the indictment (not necessary; he was a friend of Burr's and if he could have done anything for him, he would have already), and Gallatin - the one genuine friend he had in the administration - met with him often and worried about his political future. More practically, his patronage recommendations that had been ignored since 1802 suddenly were immediately tended to, and he got several judges and other fairly important governmental appointments through.

Burr wasn't particularly impressed by this - he actually voted against the Administration during the one tie breaker to come up in that session - and while he gave Chase a bit of his own medicine in how he bullied him in turn at the trial, he also conducted it so fairly and impartially (the House charges were not well constructed and Chase was acquitted, although he learned his lesson and conducted himself with significantly more decorum thereafter) that even several Federalists who had wanted to hang him when he got back to the District in November admitted that the country would be at a loss with him leaving office and his farewell address was lauded. Unsurprisingly, when he dined with Jefferson one final time after the trial to lobby for a federal appointment for himself when his term ended, he got precisely nowhere.

Edit: Slight clean up of the impeachment proceedings.