A reef aquarium can be much closer to a complete ecosystem than a freshwater aquarium.
Biological is not an issue at all. The rock alone is more than capable of providing this. In addition, live rock allows for denitrification, something almost no filter can do, and the ones that can do it cannot do it nearly as well or reliably.
The need for mechanical depends on the tank. I had a lot of filter feeders in my tank so I did not use any mechanical media in the entire system (not even as a prefilter on pumps). This did a few things. It kept any debris moving until the filter feeders ate it, did not trap debris that would have then simply rotted (increasing nitrate and phosphate), and allowed me to not have to feed anything for the filter feeders (more food means more nitrate and phosphate).
Chemical filtration is not always needed. As we all know many people do not use carbon in freshwater tanks, the same is true of reefs. Like in freshwater, water changes can usually provide all chemical filtration.