Functional elements in intergenic regions will evolve slowly because their sequence is maintained by  negative selection. In species with very large genomes, a large percentage of intergenic regions is probably junk DNA and it will evolve at the neutral rate of evolution. Junk DNA sequences are not maintained by  purifying selection but gain-of-function mutations with deleterious fitness effects can occur. 
Phylostratigraphic inference and bioinformatics methods have shown that intergenic regions can—on geological timescales—transiently evolve into open reading frame sequences that mimic those of protein coding genes, and can therefore lead to the evolution of novel protein-coding genes in a process known as de novo gene birth.