Almost all statistical models for variant calling assume some sort of independence between measurements. The duplicates (if one assumes that they arise from PCR artifact) are not independent. This lack of independence will usually lead to a breakdown of the statistical model and measures of statistical significance that are incorrect.

There are experiments where one should not make the assumption that reads that have the same start positions are PCR duplicates. In that case, using MarkDuplicates is not justified.

MarkDuplicates is important in removing PCR duplicates -- which can introduce bias in your variant calling. If you did not mark duplicates, you would risk having over-representation in your sequence of areas preferentially amplified during PCR. One way to think about it is that marking duplicates and removing them does not really have a detrimental effect on your overall depth of coverage -- but increases the quality/reliability of the areas you have covered.

There is a good discussion covered here.

And also further discussion on the Picard Main Page.