Should I tell my advisor that our paper is full of mistakes?
Your name is on the paper; you should be able to take a look at the draft, and suggest improvements.


  The paper is completely awful and full of mistakes. The thing is, my advisor will take it personally if I give comments that would put him to a bad light in front of the other co-authors. 


Okay, there are two ways to address this. You can pubically say, "This paper is awful and full of mistakes," paint your advisor in a bad light, and face his ire.

Or, you can take each of the mistakes, and tactfully offer an improvement. That is, you could privately say, "I think the paper might have a better chance of acceptance if we made these changes."

In other words, don't edit as a critic; rather, put some work into the paper as a co-author. 

There's a fine art to editing work in a way that isn't off-putting for the primary author. People tend to get defensive when their writing is critiqued – it's a very natural reaction. However, if each of your suggested changes is offered constructively, as an improvement to an initial draft, rather than an indictment of it, you stand a better chance of incurring thanks instead of wrath.

On the other hand, if the paper is so bad that it's not salvagable, then I'd suggest requesting that your name simply be removed from the author list.