No, and probably not in the long-term future either.
DNA is very good at dense information storage (about an exabyte per cubic millimeter) and (potentially) at parallel computation. It is also chemically very stable and could be used for long-term data storage.
But reading and writing information from DNA, whether by hybridization reactions or enzymatic processing, is agonizingly slow. The chemical processes involved are some 15 orders of magnitude slower than electronic gate switching.
The most plausible application of DNA drives is probably long-term storage of very (very, very) large data sets. Day to day consumer applications are not likely at all.
See my earlier answer to “What will be the economic impact of repurposing DNA for digital storage?”