A lot of consumer desktop F.D.M. printers comes with a 0.4mm nozzle. I'm looking to print fine details objects and I was considering trying to use a smaller size nozzle. But before I do so I would like to establish a list of downsides and unwanted consequences.
In addition to the answer of tbm0115; special care has to be given to the strength and stiffness of the model. Lines are thinner and thus when using the same amount of walls and infill percentage, the walls will be less stiff. This requires a higher percentage of infill or more walls to counteract this phenomenon. Obviously, this will cause the print to take longer.