Fisher will leave the club after three years, having served as Technical Advisor and Assistant Coach (Defence) since retiring at the end of the 2007/2008 season.
In addition he was Head Coach to the Munster U20s and to the Munster British & Irish Cup squad that reached the inaugural B&I Cup final.
"Laurie's experience has been hugely beneficial to Munster's squad and management," said Munster chief executive Garrett Fitzgerald.
"It's now time for Anthony to take on this responsibility and to impart his experience and great knowledge of the game to the Munster players.
"We wish him well in his new role that commences on July 1st."