Karius was at fault for Real Madrid's first and third goals as the La Liga giants ultimately eased to glory with a 3-1 victory in Kiev.
The German was distraught after the match and apologised for costing Liverpool in their first appearance on European football's biggest stage since 2007.
Jurgen Klopp said the club would rally around the 24-year-old, but Clemence - who won three European Cups with the Reds - is concerned by the long-term impact the performance could have.
"It's going to be difficult for him, that's a certainty, because wherever he goes away from home he's going to be continually reminded of it," Clemence told BBC Radio 5 Live.
"When you make two errors like that, it is difficult. He hasn't done his place any good.
"I've been there myself and it's a terrible situation. He'll be the first to hold his hands up and know he's made two horrendous errors at crucial times.
"They will stay with you for the rest of your life. People will remember them. It's watched by hundreds of millions of people around the world. And it's a final, it's amplified.
"In the biggest game there is unfortunately he's made a couple of errors that have affected the result."
Karius arrived at Anfield from Mainz in May 2016 but had only recently nailed down the number one spot, holding off competition from Simon Mignolet in the second half of the season.