Fresh leaf is the preferred tissue for DNA extraction as they contain low concentration of metabolites and polysaccharides (Jobes et al., 1995). As Sunkara Prasanthi mentioned in the above comment that dry leaves can also be used for DNA extraction but the yield and quality will be comparatively lower than the DNA isolated form young leaves.
(1) One technology called Lyophilisation, also known as Freeze-drying, can use to dry plant materials. See an attached picture of an example of Freeze-dryer below. The leaf samples are placed in the jars attached to the machine for freeze-drying. These dried plant materials can then be stored in a freezer for future use. The genomic DNA extracted from these materials is just as good as those extracted from fresh leaves.
(2) Dry leaves from plants. Cells break down from those leaves, membranes break down, DNA may all be digested from those nucleases. However, trace amount of DNA can still be found and extracted from those tissues. Many stories have been reported about isolating trace amount of DNA from dry bones or ancient foods for forensic test. The famous story is the discovery and identify of King Richard III's bones. After 500 years, scientists finally identified it from using DNA profiling and other methods. In this case, the amount of DNA isolated probably only good for such as PCR-based analysis, not for Southern-like analysis.