What I've heard from monks is that it doesn't make a big difference in the low levels, but it does in later levels. Personally on my monk, I never go far above the limit if I can help it. I do have a peacebringer, so there's not much I can do about that, but I definitely watch everything else.
Loot-wise, you can still pick up research components, gems, and jewelry, all of which weigh 0.1 each and so will not have a huge effect on your weight. I pick up those items and destroy everything else along with copper and silver. I also have one tinkerer's bag which I use for important stuff (bandages, shurikens, the occasional weighty but legitimately valuable item).
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