There is too much we do not know about the long lasting effects of GMOs. We do not know what our children or children's children will end up with because of all of the different chemicals we introduce them to. This can lead to serious birth defects in both animals and humans if we keep bringing chemicals into the body.
Let me preface this by saying I don't consider all GMOs evil or something that need to be banned outright, they serve helpful purposes like making crops grow in areas of the world they previously couldn't. That said, some of them are indeed harmful, and constant use of them can cause serious problems.
No, as far as we are aware, they cannot. There have been no studies that I am aware of that have shown that GMO's can cause birth defects in livestock or humans. I think that a lot of the fears surrounding GMO's as "frankenfood" are unfounded. Monsanto isn't a great or ethical company, but that is a different story.
The anti GMO stance of many people and many organizations pretty much falls into two camps: either anti science fear, which is fairly usual, or anti corporate fear, which is even more common in general. GMOs can't hurt people - they are not meant to be harmful and cannot be.
Is it possible that constant use of GMOs can lead to birth defects in live stock and humans? yes. It is not a proven thing as of now though. There have been some tests done and nothing has shown the risk of birth defects from the constant use. 20 years from now we may find out differently.