There are a few reasons why it's not a good idea.
The first is the suffocation factor. Unlikely, but if it is preventable, I'd always go with the voice of reason.
The second is that using a plastic bag as a barrier will trap moisture and bacteria in the mattress. Using one that's fabric based and is designed to wick moisture will help keep moisture from getting on the mattress and causing issues with mold, mildew, bacteria growing. They thrive on dark, damp environments.
Unless you are able to get a plastic tarp that's large enough to cover the entire mattress, the liquid will simply spill over the sides of the plastic bag and onto other areas of the mattress (especially when you're trying to remove it).
I'd see what you can find inexpensively at Walmart, Target, etc. I think the one we got for our son's bed was $9.99 at Target. We also use a liner underneath designed for trapping accidents. We got it at Babies R. Us for less than $10