DNA is, for the purposes of digestion, basically a protein. Proteins are broken down into amino acids by digestion, so it’s not surprising that the body would have a mechanism for it. (There’s no need to break the structure down to nucleotides, therefore.) Especially since, as Smeghead said, it will be in practically every cell of every animal and plant food that we eat. And in every single-celled organism that is inhabiting all that food.
The flip side is that the quantity of DNA is so tiny that it’s interesting the body even bothered to develop specialized enzymes to digest it. One estimate I’ve seen is that there is 6 x 10[sup]-12[/sup] grams of DNA per human cell. Of course, there are a lot of cells in the human body so eventually that could add up to a gram. You’d have to chew through a sizable quantity of food to do so. The size of DNA varies among other living creatures but this gives a ballpark for it.
Nobody actually tastes DNA, of course. Even if you’re tasting seminal fluid, the DNA is swamped by the other substances that make up the fluid portion.