It can damage modern electric and gas burners that have heat sensors built into the element/burner assembly. It reflects to much heat back up into the burner. Or it can block the sensor so you lose burner control...
That's kind of a broad statement. You have to be talking about a typical home gas range with sealed burners, I can't think of any other where that would be possible. (They do make liners for the pans under electric surface elements).
If gas is what you have I can't think of any advantage, only more trouble than it's worth going around the burners and all. The grates should lift off and cleaning the pan underneath with cleaner and a rag takes a few minutes.
You don't say what kind of stove you have and as has been pointed out it could be more of a problem than just looking tacky. There could be safety concerns. It's not a good idea to mess with how stoves are built and if new what you are doing probably would void the warranty.
Back on topic:
I do not really see any problems with lining the stove if you have a simple on/off gas stove with no electonices/electrics.
Except: you'll be creating more rubbish......
It's not so difficult to clean the top of the stove. Probably easier and quicker than lining it.
I had to think about that a little bit. A base and an acid? So I Googled it.
"How many times have you seen an ingenious solution on Pinterest touting the magical cleaning properties of baking soda and vinegar? The key is to use these two agents separately, not together — or you'll end up with nothing.