Lampard questioned the mentality of his Everton players after Sunday's 4-0 defeat to Crystal Palace in the FA Cup.
The Toffees are locked in a relegation battle, sitting just one place above the drop zone and have won four times in their last 12 games across all competitions.
Lampard replaced Rafael Benitez in January but has not been able to effect the required change at Goodison Park.
"He has to take responsibility, he's the manager" Sutton told BBC Radio 5 Live.
"The team are in an absolute mess but his language was extremely strong, questioning the side's courage."
"He talked about [how] he doesn't have a magic wand to get inside people's heads and change resilience across the whole squad. Well.. is that not his job?
"Lampard was chucking things out there and I wonder whether he will regret some of the things he's said because it can be damaging within the dressing room.
"It's not a good dressing room at this time but when you start questioning player's courage, I think there may be a problem.
"He's come out and slaughtered his players and yet he needs these players to keep him up, so what he said is very, very dangerous in my book. It's a big gamble what he said and let's see whether the players react.
"He had a difficult job to take on, most of the blame lies with [previous manager Rafa] Benitez on this one, but he hasn't made the impact which I think he expected to - and the Everton fans expected him to."
Sutton added: "I just wonder with Frank, and wondered at the time he took the job, I thought he was brave. Everton is a big club, but the club were in a real mess. This was always going to be a difficult task for him.
"His words at the weekend worried me a little bit, and they worried me because I actually thought for the first he may be doubting himself in maybe taking the job.
"My worry for Frank is if things don't work out here, where does he go from Everton? He's an intelligent man, but the way he's talking I think there's massive frustration."
