I have been told that once chilled, you should never let a beer come back to room temperature; what is the truth behind this?
If again chilled, will the flavor be the same? Or how might it change?
Temperature changes aren't great for beers, especially hoppy beers, because temperature swings of greater than 20 degrees will degrade flavor. But it takes a LOT of that to produce a noticeable effect.
Most beer probably went through a few cycles of heating and cooling before it gets to your fridge, especially in the summer. You know, maybe cold in the brewery but warmed in the truck before cooled in the warehouse then warm in the truck again before hitting a store's cooler.
It'd be hard to notice anything. Hoppy beers might have their big aromas fade a bit or get a less smooth bitterness. Anecdotally, some of the smoother DIPA style beers with big floral or citrus hop smells/tastes might degrade into more biting bitter territory.
Big beers made for aging will generally be forgiving but heating and cooling them a lot might make them taste flatter or cardboardy. But again, we're talking several upon several times before anything noticeable happens.
Long story short...it's probably not going to do anything you're going to notice chilling unopened beers back down from room temperature.