No, DNA replication is not energetically expensive for eukaryotic cells. Most of the energy in a cell is spent on protein synthesis (i.e. gene expression), so regulatory mechanisms to suppress or promote DNA transcription are very honed and fine-tuned. These mechanism are also co-opted into fighting DNA parasites but they can't be used to edit away junk DNA completely.
However, for bacterial cells (which are much smaller than eukariotic cells) DNA replication is a significant burden and so they have very little DNA junk. That holds true for intracellular symbiotes like mitochondria and chloroplasts - their genomes are under (some) pressure to get simpler and smaller.