Here's why Burn Notice's Fiona stopped using her Irish accent after the pilot episode. Burn Notice was a slick spy series that arrived on the USA Network in 2007. The show follows Jeffery Donovan (Sicario) as "burned" spy Michael Westen, who finds himself stranded in Miami after being labeled untrustworthy. Burn Notice follows Michael as he investigates who burned him while taking on assorted odd jobs in Miami. The show is narrated in a unique second person manner, with Michael talking directly to the audience.
Michael has a few allies in Miami, including his mother Madeline (Sharon Gless), ex-NAVY Seal Sam Axe (Bruce Campbell, The Evil Dead) and ex-flame Fiona Glenanne (Gabrielle Anwar). Burn Notice would run for seven seasons in total and made for a highly entertaining action series. The show also received a spinoff TV movie with Burn Notice: The Fall Of Sam Axe, a prequel that detailed how Sam ended up in Miami; Jeffrey Donovan also directed this movie.
The on-again, off-again dynamic between Michael and Fiona made up a big part of Burn Notice, and while they received a well-earned happy ending in the finale, it was a long road to get there. Fiona's backstory is that she was a member of the IRA and became involved with Michael while he was working undercover in Dublin. Their romance was far from smooth, however, and came to an abrupt end when his cover was blown and he left without saying goodbye.
Fiona helps Michael out in the Burn Notice pilot once he's stuck in Miami, and throughout season 1 tries to reignite their relationship. She also spoke with a clear Irish accent in this first episode, something which was dropped for the rest of the series, aside from occasional episodes. The main reason is that Fiona is trying to remain low-key during her time in Miami, and she adopts an American accent to fit in. Her original accent will slip out if she's talking to other Irish characters but despite the show ending with her back in Ireland with Michael and Charlie, she still speaks with an American accent.
The explanation of using an American accent to fit in is the same one given for Elizabeth Olsen's Scarlet Witch in Avengers: Infinity War, who originally spoke with a heavy Eastern European accent. Another likely explanation is that an Irish accent is tricky to maintain, so Gabrielle Anwar was probably happy to switch to an American accent for the remainder of Burn Notice instead.