There is an important distinction that needs to made, which is germ line cells vs somatic cells. Germ line cells form into the gametes, which contain the DNA that gets passed on to their offspring. On the other hand, somatic cell DNA doesn't get passed on (e.g. a random lung cell is a somatic cell).

Germ line cell DNA also doesn't mutate as readily as somatic cell DNA. However, a bad diet can lead to epigenetic changes (that influences gene activity and expression without actual changes in the DNA sequence) in germ line DNA, which can affect the offspring.

Of course, it's a lot more complex than I just described.