Apparently 80% of people in the USA think so, according to a Washington Post article that's been all over Facebook in the last few days. That is, 80% of those polled in the regular Food Demand Survey (by Oklahoma State University's Department of Agricultural Economics) agreed with the proposition that all food containing DNA should be labelled. (To put this in context, there is currently a heated debate in the US - driven by those opposing the incorporation of material from genetically-modified organisms into the food chain - over whether such foods should be labelled as such.)
Now, you could argue that the question was poorly worded. There's been a certain amount of skepticism that those in agreement with the DNA proposition could be so high - after all, anything with whole cells in it will definitely contain DNA, & there'll probably be traces in most other foods, apart from very highly processed foodstuffs like refined oils and sugars. And salt. Perhaps they thought they were talking about foods from genetically-modified sources, as opposed to 'natural' foods (more on that later)?
Perhaps. But there was also a question on that.
The author of the Post article suggested that the poll results were the outcome of "the insection between scientific ignorance and political ignorance", and went on to say that perhaps many of those polled "don't really understand what DNA is, and don't realise that it is contained in almost all food."
This is close to the 'information deficit model': the one that argues that if 'laymen' are given all the information on the scientific issue du jour, that they will change their minds & accept the scientific perspective. However, this ain't necessarily so. As that debate around labelling of GM foods shows, there are far more factors in play than simple (lack of) scientific understanding: do people feel that their voices have been heard by those making the decisions. Do they have particular religious beliefs that affect their attitudes? How much of their feelings on the subject are shaped by personal ethical perspectives, or individuals' experiences? This means that those communicating about science need to be aware of these perspectives and frame their communication accordingly, with an eye to real engagement rather than simply throwing information at people.
In New Zealand these issues & others were canvassed by the Royal Commission into Genetic Modification, back in 2000. This was a good example of the sort of meaningful engagement with the public that needs to become more widespread, although looking at how these questions are addressed in schools could also be interesting. I know that back in the early 2000s, we found that a small proportion of new first-year students were aware that all living things - & not just GMOs - contained DNA. A much, much, much smaller proportion than in the US survey! So at that level, maybe we're doing something right :)
Coming back to the 'natural' vs GMO foods: geneticist Kevin Folta has noted that modern GM techniques give far more control, in terms of known genetic & phenotypic outcomes, than hybridisation or mutational breeding (& that genes can and do move between species without human intervention). There's a useful graphic, comparing the outcomes of the different techniques, here.
Oh, & the Washington Post wrote a rather tongue-in-cheek mock-up of what a food label might look like, if public opinion results in such labelling becoming mandatory:
WARNING: This product contains deoxyribonucleic acid (DNA). The Surgeon General has determined that DNA is linked to a variety of diseases in both animals and humans. In some configurations, it is a risk factor for cancer and heart disease. Pregnant women are at very high risk of passing on DNA to their children.
EDIT: For other comments, try Kavin Senapathy's post, and also this thoughtful piece on whether the question was actually inappropriate in its context, by Ben Lillie (and thanks for the heads-up on Ben's post, Grant).