I wouldn't add any biological filtration as the microbes and critters growing on/in the sand and rock should do that very effectively. Other forms of filtration such as protein skimmers, granular activated carbon (GAC), granular ferric oxide hydroxide (GFO), etc. can all be very useful for removing certain substances from the water, improving water quality.
If we compare our aquariums to nature we see that most tanks have a far, far higher density of fish than in nature, and hence a higher production of fish waste than in a comparable area in nature and larger need for nutrient processing as a result. Just to give a ball-park idea of realistic fish densities compared to natural reefs: a recent study in PNAS found that on healthy, essentially unfished reefs in the Indian ocean the biomass of fish is about 1000-1500 kg/hectare of reef. Scaling that down to an aquarium, that would be about 4-6 fish about 3-3.5" long in a 120 gal aquarium (or maybe one 5-6" fish). Most folks keep a lot more fish than that.
The far bigger issue, however, is the relatively short residence time of water on coral reefs as compared to in our tanks. On a normal reef the water residence time is typically about 2-12 hrs, depending on the reef and wind, primarily, which drives waves and water over the reef. This means that reefs in nature get essentially a 100% water change about 2-12 times per day. Excesses of waste or other material are quickly washed away and ultimately remineralized elsewhere. Most reef tanks get, what, maybe a 10% water change once per week to once per month? Excess waste material and other nasty substances that are simply washed away from the reef and processed elsewhere can easily and rapidly build up in a reef tank. As a result it is often necessary to use filtration equipment (skimmers, GAC, GFO, etc.) to remove these excess wastes to keep the tank running well. Exactly what sort of filtration is required or at least useful varies from tank to tank, depending on the individual situation.
Reef tanks are somewhat of a hybrid between natural reefs and wastewater treatment plants. In many ways they operate similarly to natural reefs, but we also put dramatically higher demands on the natural filtration capacity by (usually) stocking higher densities of fish than are natural, and by performing dramatically less in terms of water changes. For these reasons, we often need equipment to help us achieve healthy conditions.
cj