Hughes et al. (2005) report on the nature of Y chromosome genomic differences between humans and chimpanzees. The paper is a test of the hypothesis popularized by Bryan Sykes (in his book Adam's Curse: The Science That Reveals Our Genetic Destiny): namely, that in a few million years, the human Y chromosome will disappear entirely because its genes will have become entirely inactive or subsumed on other chromosomes.
The short answer is: not gonna happen.
The long answer has some interesting twists. As it turns out, humans appear to have conserved all of the functional Y chromosome genes that occurred in the human-chimpanzee common ancestor. The paper proposes that this is evidence for a stronger effect of purifying selection in humans than might have been assumed. They were able to confirm this by comparing coding divergence with intron divergence; the coding region sequence divergence between species was significantly low. That leaves an interesting evolutionary question: how strong was purifying selection compared to other chromosomes, and could it have affected standing Y chromosome variation during human evolution? Unknown.
But these genes are ordered very differently on the Y chromosome than the equivalent genes in chimpanzees. In chimps, all the X-degenerate genes are together in one region. In humans, they have been scattered around onto both arms of the Y and intermingled with other sequences.
The study is only of a subset of the Y chromosome: the X-degenerate regions, or the parts that contain genes with X chromosome analogs, but only analogs that are significantly divergent from their X chromosome equivalents. So it does not represent a final answer on many of these questions, but it constitutes what may be the most important source of Y-unique evolution.
Although humans retain all of the functional genes of the common ancestor of the two species, chimpanzees have actually lost several. The authors propose a hypothesis to explain this loss:
Why have X-degenerate genes decayed in the chimpanzee lineage but not in the human lineage? We speculate that X-degenerate gene decay in the chimpanzee lineage may be a by-product of strong positive selection focused elsewhere on the Y chromosome, through a process known as genetic hitchhiking. Because the Y chromosome does not participate in sexual recombination with a chromosome homologue, natural selection acts on the chromosome as a unit. Deleterious mutations in some Y-linked genes can be carried along, even to the point of fixation in a population, by physical linkage to strongly beneficial mutations in other Y-linked genes. In addition to their X-degenerate genes, primate Y chromosomes contain many families of ampliconic genes, which have testes-restricted expression patterns and critical functions in sperm production. Because of this central role in spermatogenesis, the Y chromosome's ampliconic genes may be subject to powerful selective pressures, especially in species such as chimpanzees where females usually mate with multiple males, the sperm of which then compete for a limited number of oocytes. During chimpanzee evolution some X-degenerate genes may have been casualties of selective forces directed at the Y chromosome's ampliconic genes--forces that were not as intense during the evolution of our less promiscuous species (Hughes et al. 2005, references omitted).
Could be true. On the other hand, the fixation of a mutation that deactivates a gene is pretty drastic, even if it is hitchhiking with another favorable variant. It certainly implies that the deactivated genes have little adaptive importance to start with. So this reduction in adaptive importance for the deactivated genes must still be explained. The physical difference between ampliconic and X-degenerate genes may have something to do with it, but it can't be the whole story.
Hughes JF, et al. 2005. Conservation of Y-linked genes during human evolution revealed by comparative sequencing in chimpanzee. Nature 437:100-103. Full text (subscription required)