Mormons believe that a small group of Hebrews, led by a guy named Lehi, came to North America and that modern Native Americans are their descendants (specifically, descendants of Lehi's son Laman). Christ supposedly visited them after he did his shtick in Palestine.
The archaeological, genetic, and linguistic evidence is uniformly against this theory. Genetic analysis universally shows an East Asian origin of North Americans, without any trace of Middle Eastern haplotypes. Archaeological evidence shows no sign of any massive civilizations of Hebrews--or any sign of ancient Hebrew technology, culture, religion, language, nada.
In this case, the lack of any evidence is so incredibly overwhelming, that I think it's acceptable to say that absence of evidence is evidence of absence. Any civilization of the sort described in the Book of Mormon would not only leave its own evidence behind (ruins, artifacts, etc), but also would have a massive impact on other cultures; items traded to them, technology, cultural forms and styles, language that gets adopted and assimilated into other languages, etc.
Two examples: The Book of Mormons explicitly describes the use of steel and wheeled chariots. These are two technologies which do not appear anywhere in the archaeological record (to say nothing of horses, which were unknown to the continent). These are also technologies that we would not just see at the Nephites' and Lamanites' sites, but also at the sites of peoples they traded with. Even if the Nephites and Lamanites didn't trade with anyone, and died out without leaving any cultural descendants, other peoples in the region would have identified such technologies as useful and ripped it off or figured it out themselves. We would expect to see that in the archaeological record. But we don't see any of that, anywhere, at any time--which indicates rather strongly that there's nothing there to find.
It is therefore accurate to say that at this point, we can consider the "Book of Mormon" hypothesis to be scientifically falsified.
The problem here is that unlike the Bible, which can be (and is) taken metaphorically, the historicity of the Book of Mormon is core to the LDS religion. If it's false, the entire belief structure falls apart, as it's dependent on the accuracy of Joseph Smith's "translation" and the veracity of his communications with God.
In the very near future, the Mormon denominations (not just LDS, but Community of Christ and the various smaller sects) are going to have to either make a radical ideological shift, or make an extreme anti-science turn. I find the third alternative, which is that they acknowledge that their religion is the made-up fantasy of a 19th-century cult leader, to be rather unlikely.
) of Mormonism isn't all that different from any of the major religions as is seen; a guy mumbling stuff into his hat, but these older religions are shrouded in the murk of their beginnings being selected and edited by centuries of workers refining the themes for something palatable to the casual observer.