Diagnosis
That's almost certainly worn-out dryer drum support rollers. They're plastic wheels with a rubberized outer ring that the heavy drum rides on. When the rubber wears flat or the bearings seize, you get a loud rhythmic scraping or squealing.
How to Fix It
You'll need to replace the rollers. Unplug the dryer. Access is usually from the front: remove the lint filter housing, then the top panel (pops forward after releasing clips or screws at the back), then the front drum panel. The rollers are on posts at the rear of the drum cavity. Spin them by hand—if they don't spin freely or are visibly worn flat, replace them. It's a common repair kit.
If That Didn't Work
If the noise is more of a constant humming or rumbling, it could be the idler pulley (which tensions the drive belt) is failing. It has a bearing that goes bad. If it's a high-pitched squeal that changes with drum speed, the glides or felt seals at the front of the drum might be worn out and the metal drum is rubbing on the front bulkhead.
Heads Up
While you're in there, check the condition of the drive belt. If it's frayed or cracked, replace it at the same time—it's right there and saves you from taking it apart again in six months.