I think you will still need a filter regardless unless the tank is very heavily planted and has a very small bioload.
3 loaches, is this clown loaches or khuli loaches?
But nonetheless, a filter is still needed to keep the water parameters at a good level, and cleaning water of debris and particles as well as giving flow to the tank water.
So, imho, it makes more sense to keep a filter running in your tank rather than not.
I would answer your direct question with "no," you do not need a filter provided the tank is biologically balanced. But, having said that, and as someone who has experimented with a filter-less tank, I would recommend a filter even if it is simply a sponge filter. The reasons Ch4rlie gives are why I too would recommend a filter.
I kept a breeding pair of large fantail goldfish in a filterless heavily planted tank for many years (til some malicious neighbour tossed poison into the tank), they bred regularly and i havested many baby fantails from the tank.
I don't have the numbers, and of course they vary from case to case, but provided the aquarium is in balance (i.e., not stocked beyond its natural capacity) you will find more than sufficient oxygen. Of course, issues can arise that may lower the oxygen, but then this can occur in any aquarium with filters too.