No, they don't disprove it. However the Alcubierre metric, which underlies the warp drive stuff, requires something people refer to as 'exotic matter' which in this case means matter with negative energy density. Such matter violates many assumptions, and would normally be considered not to be possible (indeed my memory of Alcubierre's original paper is that he said more-or-less this).
One of the reasons people assume that such matter is not possible is that if it were then very bad things follow, such as causality violation (ie being able to travel into your own past) and, really, the collapse of most of physics. [The following is opinion, not answer.] Of course, other groups of people want it to be possible either because they want warp drives to be possible and do not understand the consequences of them, or because they do understand the consequences and are happy with them.