The objects which Nate examines give background information on the plot, specifically Henry Avery's background, along with information about Libertalia and the treasure hunt. If you want to complete the game properly, I'd suggest looking out for these things, although it is very time consuming.
As far as I know, you don't necessarily get any trophies by examining all of the items.
You do get a trophy for finding all 'Strange Relics', which relates to other Naughty Dog Games. For example, the Strange Fruit Relic relates to Crash Bandicoot.
However, by collecting all journal notes in the game, you do get awarded a trophy, but it can be quite tedious. You would also need to know where to look.