A staff member at one venue told the BBC they wanted to ensure the experience was not spoiled for younger children on their first cinema trip.
Videos showing young people dressing up in suits to watch the movie in large groups and cheering loudly during the screening have gone viral on TikTok, racking up millions of views in some cases.
The teenagers filmed themselves copying the characteristic steepled fingers of the movie's supervillain Felonious Gru and celebrated that "the five year wait is over" - referring to the time since the franchise's last instalment, Despicable Me 3.
Movie studio Universal Pictures endorsed the trend, saying on Twitter: "to everyone showing up to @Minions in suits: we see you and we love you".
But cinemas have been less appreciative, with several complaining about the behaviour of the young fans drawn to the movie by TikTok.
Mallard Cinema manager Daniel Phillips-Smith: "It's been absolutely heartbreaking. We've had families who won't even go back into the screen when we've tried to sort it out, families leaving before the film has even started, and of course the children have been in tears."
"We were trying to talk to people on the way in after a while, but that wasn't really getting anywhere. So we had to just decide to put a stop to it and say thanks very much, but we're not doing the whole TikTok trend thing," a staff member said.
"We had people coming in with children for their first cinema trip ever and we wanted to protect that really."
The manager of the Vue Cinema in Worcester told the local Worcester News they had to refund Â£1,300 of tickets on Sunday because a group of young people had caused so many complaints.
Images on social media showed several Odeon cinemas displaying signs warning customers: "Due to recent disturbances following the #GentleMinions trend, any group of guests in formal attire will be refused entry for showings of Minions: The Rise of Gru."