I have a Muji utility bicycle with a 3-speed internal gear hub which I got second hand around 5 years ago. The LBS which I bought the bike from has since closed down and I haven't managed to find another bike shop which does maintenance on this kind of bike (the last one I visited just told me if my bike broke down I should get a new one). I was wondering how complicated it would be to do maintenance by myself and if anyone has experience with internal gear hubs?
My experience as a bicycle mechanic in the Netherlands: don't mess with it.
If it is an Shimano, it will probably run >100.000 km. If it is a Sturmey Archer, SRAM or any other well known brand, it will mostly do >50.000 km as well.
If there is a small hole for lubricant, only do this once a year or so, with some thin grease, but not oil. Oil is to thin.
If it doesn't shift well, first look at the cable. You probably only need to adjust the tightness of the cable. If this doesn't work, de-attach the cable. Every hub must work correct in the smalles or biggest gear without a cable attached. If this still doesn't work, the hub is broken. (Happens almost never.) If a 3-gear hub is broken, it is easy to take apart and build together again. And finding which gear or bearing is broken isn't difficult either. Finding a bearing balls is easy to, but finding a spare gear is probably not so easy.
Main point: leave it as it is.
In my experience with various internal gear hubs on Dutch bikes, I found there's no maintenance necessary: I've never heard of them breaking down because of lack of maintenance. Here's a blog post that concurs.
My current Gazelle bike manual also does not explicitly mention any maintenance on the hub gears, only how to adjust them.