Shaw was handed his first start of the campaign against CSKA Moscow on Tuesday evening, making the most of his rare chance by producing a solid display down the flank in the Red Devils' 2-1 win.
The England international, whose only other appearances have come from the bench in the EFL Cup, has seen his fitness levels called into question in the past.
Scholes believes that it is important Mourinho selects Shaw more regularly, comparing him to former teammate Wayne Rooney.
"It's difficult to say as we don't see what goes on behind the scenes. I don't think he lives his life as clean as he possibly should do," he told BT Sport. "With his body shape, he looks like a player who needs to be playing every week.
"We had Rooney, who was a player if he missed a couple of games you knew, he wasn't looking after himself. He needs to be playing all the time, Saturday, Tuesday, whatever it may be. With Luke Shaw, is he going to get a run of 10 games in the team?
"I'm not sure, the problem he has got is that Ashley Young has been brilliant at left wing-back. I hope there is a future for him. A couple of years ago, I thought we were signing the best left-back in the world."