Cough syrup with only DXM in it won't hurt your liver unless taken massively over a period of something like 10 + years. Even then, the damage would be mainly due to the syrup part. Additionally, I have not seen any evidence of this even, it's merely my own speculation.
Liver damage happens when you take the wrong kind of DXM.

I agree that weed is better! Man, but I havent had a hook up for 5 - 6 months. After that period of time, non-voluntarily youll be literally the next in line to pick up some DXM.