I'm picking up on a couple of questions raised in this thread. As we can see from the posts, this is a subject on which there is varying opinion among very experienced aquarists.
In a healthy established tank, there is a natural water movement just as there is in nature. Water in the tank is drawn down into the substrate where it is slightly warmed (due to the biological processes occurring in the substrate involving bacteria, organics, plant roots, oxygen) and then returns up into the tank where is cools again. In other words, a top-to-bottom circular movement. However, as Mikaila touched on, circulating water is very important for uniformly heating the aquarium. Warm water rises, but in especially larger tanks in length, it will not likely get around the tank without some water movement.
Turning off the filter for extended periods can be dangerous, depending upon the filter design. Aerobic nitrifying bacteria colonize the filter media, and they need oxygen which comes via the water flow. Especially in a canister filter that is closed from the outside air, stopping the water flow for an extended period (such as during the night as someone mentioned) will cause the bacteria to die from suffocation. Then when the filter is started again, all these dead bacteria are moved into the aquarium. No different than having dead fish lying about. This is why during a long power outage, canisters should be disconnected (turned off) and then rinsed out before being started up. Of course, the effect of this also depends upon the initial bioload in the aquarium. Tanks that are densely stocked, or overstocked, are much more likely to have serious problems fast.
Nutrients are carried to plants via water movement. As mentioned above, water is naturally drawn down through the substrate and without this the plant roots would be hindered in assimilating nutrients. Nutrients in the substrate--be it soil, sand, or gravel--must enter the water in order to be assimilated by roots or leaves in those species that use leaf assimilation more. Anaerobic bacteria also play a role in this, as a source of nitrates.
Water movement also keeps particles from settling on the plant leaves; a buildup can prevent the natural plant respiration through the leaves, causing death of the leaf and eventually the plant.
I too have experimented with a filter-less tank. It was a small tank (10g) with lots of plants and very few fish--and lots of snails (Malaysian livebearing). After several months, I added a sponge filter as I was not happy with the situation. I believe things are better now.
It is one thing for experienced aquarists like those participating in this thread to "experiment" with this or that, because they are more likely to understand the ramifications and prevent trouble. But for the majority, and certainly those new to the hobby, filters that are not too powerful for the good of the plants and fish are probably a very good idea. And, after my 20+ years I still have filters in all my tanks, and see no reason not to--just in case.