I think the most disadvantages are more general multiplex related, and not so specially for Luminex. Since the Luminex system is mostly the readout-system.
For antibody based assays one of the obvious things is that every target is measured in the same system, so the same sample dilution, same buffer system etc. This could result in compromises in for example dilutions/sensitivity.
Next thing is that secondary antibodies might cross react in a multiplex assay, resulting in false positive reactions. Careful selection of ingredients is essential!
In nucleic acid assays I think the biggest problem is in designing multiplex PCR reactions. That is always a challenge. Although there are different assay formats that might circumvent this problem depending in the targets.