No archaeological evidence
Mainstream (non-Mormon) Mesoamerican scholars see no evidence of the presence or influence of Old World cultures in pre-Columbian Mesoamerica, nor has material evidence been found that would indicate ANY contact between Mesoamerica and Old World cultures. This has been well documented http://en.wikipedia.org/wiki/Archaeology_and_the_Book_of_Mormon
No DNA evidence
Scientists have analyzed the DNA of 1,164 Mesoamericans (see table below). They have focused on mitochondrial DNA because it is easy to analyze and it is transferred down maternal lines, thus avoiding admixture problems (males dominated early colonizing parties after Columbus).

n= number of individuals tested in each population
(from http://simonsoutherton.blogspot.com/2011/12/where-are-lamanites-in-mesoamerica.html#comment-form)
The A, B, C and D lineages are derived from Asia and are not found in the Middle East. Eleven Mesoamericans had a mtDNA lineage that didn’t originate in Asia. Three had African L lineages, and three were not fully characterised as the studies were performed over 15 years ago using out-dated methodology. The author’s believed that these lineages may well be from the 4 major Asian lineages.
The remaining five lineages are all found in Western European populations. Exact mtDNA matches have been found in individuals from Spain, Portugal and Poland. Clearly they derive from the small number of European women who migrated to the Americas after Columbus.
You would think that by shrinking the geography that the Lehite influence in that smaller region would be easier to find. Instead we find that 100.0% of Mesoamerican DNA is derived from Asia.
No Lehites
Mesoamerica was selected by limited geography apologists as the most probable setting for the Book of Mormon due to the impressive civilizations that are found there. Science has now conclusively shown that Mesoamerican cultures and the people living in those cultures are not connected with the Old World.
Its time the Mormon Church faced facts and stopped imposing their myths on innocent native people in Mesoamerica. These people are not descended from father Lehi. http://www.ldschurchnews.com/articles/61817/Quetzaltenango-Guatemala-Temple-2400-youth-participate-in-cultural-celebration.html
They have much more ancient, meaningful and real connections with the New World.