You must tear it off, or the mattress will mildew.
You can tear it off but you should buy a cover instead.
You can get a mattress cover. That will be better.
Yes, you need to. Otherwise, the mattress will be unbreathable and uncomfortable.
It's not a "need" issue. It's a "must" issue. Why? The most important reason is that the film is not breathable. And it affects the comfortableness of the mattress and let alone its hygroscopicity. If you are afraid the mattress gets dirty, you can do as what the former answer said--buy a mattress cover. It's not expensive.
