No. Adding additional sex chromosomes won't increase genetic diversity.
The benefit of sexual reproduction lies in the fact that a population can create offspring with any combination of chromosomes that are present in the population. An asexually reproducing population, on the other hand, can only create offspring with a set of chromosomes that is already present in the population.
The important thing to keep in mind is that the sex chromosome is only one of a large number of chromosomes. Just like offspring receive one of two sex chromosomes from each parent, the offspring will receive one of each other chromosome from each parent, as well. Which chromosome the offspring receives from each parent for non-sex chromosomes is independent of which sex chromosome they receive from that parent. Two parents with the same AB heterozygous chromosomes for all chromosomes but their sex chromosomes, for example, can produce offspring that are any combination of homozygous (either AA or AB) or heterozygous for all of their non-sex chromosomes. Adding a third sex chromosome won't increase the size of your potential gene pool, since it's already 'all of the genes'.
On the other hand, if three individuals with three different sex chromosomes must be present in order to reproduce, your species is more likely to go extinct at small populations, since if all of any of your sexes die off, your species will perish. This is compounded by the fact that, should a genetic defect exist on one of the sex chromosomes, the population of individuals with that sex chromosome will be smaller compared the portion of the population with each sex chromosome in a two-sexed species. (i.e. 20 individuals rather than 30 in a total population of 60.) This will make it more difficult to breed out that defect, since the population of chromosomes for that specific sex chromosome will be smaller. A defect will also be more likely to wipe out your species because it makes up a larger portion of a smaller pool of individuals.
On the other hand, a sexually reproductive species with only one chromosome-based sex will be more robust in the face of extinction than a two-sexed species, since any two individuals in the population will potentially be able to mate and produce offspring, rather than requiring a male and a female.