The initial match was abandoned after Besiktas head coach Senol Gunes was struck on the head by an object thrown from the stands at Fener's Ulker Stadium.
Gunes required hospital treatment and received five stitches after the incident in the 57th minute, which prompted the match to be halted at 0-0 and 2-2 on aggregate.
The Turkish Football Federation announced on Wednesday the rest of the game would be completed behind closed doors on May 3.
But in a statement released on Thursday, Besiktas said the governing body's decision was met with "sadness and regret" and announced they would be boycotting the match.
“Even though millions saw clearly that Besiktas were the only victimised side, the Turkish Football Federation ruled that our Turkish Cup semi-final second leg against Fenerbahce should continue from where it was abandoned," it read.
"As a club who adhere to fair-play at all times, Beşiktaş JK have decided not to take part in the said Cup game, scheduled for Thursday, 3 May 2018.
"We believe that this will be the most dignified stand for the great Besiktas supporters to have their voice heard.
"For 115 years we've always been on the field, winning or losing. We will not there on Thursday, May 3, 2018 to accept the loss at the Masada.
"We expect our supporters not to allow the tension to continue, to keep fighting for a third consecutive championship and supporting the team."
Besiktas are currently second in the Super Lig, just one point behind Galatasaray - their next opponents.