It's tremendously hard to know exactly how consumption of genetically modified organisms will impact human health in the years to come. The truth is that we haven't been consuming GM plants (namely corn and soybean) for long enough to adequately study their effect. New research provides some level of insight, however, toward what might happen, physiologically, if we continue to include GM organisms in our diet.
In a research lab at the Norwegian School of Veternary Science, rats were fed either GM or wild-type corn over the course of 90 days. There were no appreciable health benefits to consuming genetically modified corn, but there was one noticeable concern - body mass. Rats fed the same amount of GM corn were differentially fatter than those who consumed the wild type. The same was true for rats consuming salmon that had eaten GM corn (a good example of how GMOs can make their way up the food chain).
So what does this mean for humans? Possibly nothing. While rats do serve as a good model organism, they are certainly not the end all, and it is possible that this phenomenon is isolated to rodents. It is also possible that GM corn is having an impact on human obesity. The safest bet is to continue eating the whole wild foods that we know and love.
Genetically Modified Foods May Be Why You're Fat from FastCo Exist