Is the genome in the seeds of the plant turned off? That is, does the DNA in the seed not self-replicate? My second question is, what are the cells feeding on inside the seed before it is planted?
DNA in cells in general replicates when cell division occurs (unless it is a cell that contains many nuclei or multiple copies of its own genome, in which case DNA would replicate as copies are made. Multinucleate cells include things like the mycelium of fungi, and here is an example of a bacterium with thousands of copies of its own genome).
As long as a plant embryo is developing, then the cells are dividing and the DNA is replicated. You might be thinking of the stage of seed dormancy; insofar as during this stage the embryo is no longer growing, cells aren't dividing and therefore DNA is indeed not replicated. I don't know whether we could say the genome is "turned off" however, DNA's role isn't just being replicated, more importantly its role is to be transcribed into RNA which leads to the proteins the cell needs to function being created. I couldn't find whether DNA transcription happens during cell dormancy but it's plausible enough that it is at least slowed down as well. The cells presumably don't have much to do during this period and you wouldn't want them to do much, to conserve energy.
Seeds contain nutrients that can feed the embryo until it can feed itself (for plants that is the moment when they germinate enough to photosynthesize). That is where flour comes from!