Former Scotland forward Boyd believes "alarm bells" will be ringing among the Celtic board over the fans' reaction to manager Ronny Deila and a Scottish Cup semi-final win would only "paper over the cracks".
Less than 15,000 fans turned up at Parkhead on Sunday to watch Celtic beat Morton 3-0 in the quarter-finals, hours before they were drawn out to face Rangers in the last four.
The Kilmarnock striker, promoting the game for the sponsors, said: "I don't think whatever happens in this game is really going to make a difference to whatever Celtic are going to do in the future.
"When you see there is only 14-15,000 turning up, it's not a good sign for the club and I fully expect an overhaul there in the summer, whether it's players leaving because of the amount they have got, or an overhaul of the management staff.
"In football that's usually what happens, it's the manager that goes.
"I can't remember the last time I read a positive story coming out of Lennoxtown or Celtic Park. It doesn't look good in terms of what's happening in the future. Something is going to have to change in there.
Celtic's Gary Mackay-Steven (R) in action with Greenock Morton's Lee Kilday
Image credit: Reuters
"They are top of the league, but the Celtic fans want to run away with everything and win everything and it's another season where they have not been able to do that.
"All credit to Aberdeen, they have hung in there but with the money Celtic are spending and the players they have got, you would have fully expected them to pull away and do the exact same Rangers have done in the league below.
"I don't think this game will change a lot in what the Celtic hierarchy will do in the future.
"If they win it might paper over the cracks for a few weeks but the Celtic fans, the ones that I know anyway, are not happy with the way they have been playing, they are not happy with things that are going on connected with the club.
"And I don't expect a one-off game to change that. A lot of them have made their mind up already.
"And for the Celtic board, if Celtic are playing in front of 15,000 in the quarter-finals of a cup, I think the alarm bells will be ringing now."
Boyd was an unused substitute when Rangers lost 2-0 to Celtic in last season's League Cup semi-finals without troubling Craig Gordon, but he believes his former club might have an edge for the Hampden fixture in mid-April, especially after their 4-0 quarter-final triumph against Dundee.
Boyd feels manager Mark Warburton has instilled a togetherness and a system akin to Claudio Ranieri's impact at Barclays Premier League leaders Leicester.
Celtic manager Ronny Deila has been heavily criticised.
Image credit: Reuters
"I expect Rangers to come here full of belief," the 32-year-old said. "They believe they can do anything right now and rightly so because when you look at the way they performed on Saturday it was excellent from the word go.
"It can go a long way in football, as we are seeing down the road with Leicester, when you have a tight-knit group and there's not a lot of change. And the ones that do come in know exactly that they have to do. That can take you a long way.
"It's going to be close but I am veering towards Rangers because they are a tight-knit group.
"The ones with a steady team, with no chopping and changing, you go to training all week fully expecting to play. Whereas you look at Celtic, they could field three or four teams."