I'd expect about every nation knew the war was going on, but were there any nations that were largely unaffected in any way throughout the war?
There were neutral parties all over the globe, and it would vary heavily based on that. For example, Ireland was officially neutral, but due to its proximity to Britain, it would from time to time be hit by mistake. For example, Dublin was hit: http://en.wikipedia.org/wiki/Bombing_of_Dublin_in_World_War_II (I regret to have to use Wikipedia, but they reference several contemporary newspaper accounts). There were instances of cooperation (as documented in the Cranborne Report, though I am not able to find the primary source right this second).
Iceland was officially neutral, but was invaded (or garrisoned, depending on how you want to word it) by Britain in 1940. http://www.bbc.co.uk/history/ww2peopleswar/timeline/factfiles/nonflash/a1126496.shtml. It is pretty obvious that Iceland's position in the Atlantic made it very important. (Iceland went independent when Denmark fell.)
There were a lot of countries that were not affected much, but I think every nation on some level did something. Switzerland was neutral, but prepared for invasion (that never came). South America was probably the least affected, but there were apparently some impacts there (I'm not at all familiar with the history there, though).
I think youre going to need to define a timeframe because every nation on earth has been affected by the results, maybe an uncontacted tribe in south america?
Interesting question.
I wonder also what effect WW2 had on completely neutral countries, eg how it effected their post war economy etc
Mozambique and Angola were neutral and far away from the frontline.
http://en.wikipedia.org/wiki/Participants_in_World_War_II#Argentina
Argentina was almost unaffected. It declared war on Germany a month before VE. Having said that, it was impacted to the extent that it exported goods to warring nations.
Afghanistan came to mind as a possible candidate, so I looked it up:
http://stonebooks.com/history/afghanistan.shtml
TL;DR. Afghanistan tilted toward Germany until mid-1941 (mostly for anti-British reasons). Once the Soviet Union and the British Empire were both at war with Germany, Afghanistan maintained neutrality but never cut diplomatic relations with Germany.