No. Furthermore, the line "once you change someone's DNA" contains a wealth of misconceptions.
Once you have your potato/tomato/whatever, the DNA does not change during the life of that individual specimen. The genetic modifications change the seed before the individual specimen grows, and do not change the individual specimen during its life.
As far as genetic modifications go anyway, each seed inherently has different DNA from its parents. "Normal" breeding has given us a diversity of plants and animals which are so massively different from the originals as to be virtually unrecognisable as the same species in many cases. (If you didn't know better, would you think a dachshund and a husky were the same species?) And whilst we might think of it as "normal" today, when the techniques around selective breeding were first developed, they in turn were seen as highly unnatural, particularly the practise of breeding an individual with a parent or sibling.
This still relies on normal rates of DNA mutation within the same species. More interestingly though, DNA can also be picked up from other sources. This is identical to how scientists insert or replace genetic material for GMO - indeed, GM techniques to carry this out were founded on understanding how this takes place in nature.
This is where objection to GMO falls down. Sure, the result may or may not be healthy. However you are already using/eating/breathing pollen from individual specimens which have carried out gene transfers with other species. The difference between the natural version and what happens in the lab is only the same as the difference between natural breeding and selective breeding - it's simply that someone is deliberately choosing which attributes to pass on to the next generation, using tools which already exist within the organism's DNA.