I think you can customize it to completely get rid of them but i just have it set to not show the same message twice within 5 seconds or something like that so you still see a message, just not the same one over and over.
Without any addons, the following script commands work:
/script UIErrorsFrame:Hide() - completely hides error frame. Since the yellow quest loot message is linked to the same frame, it hides those too. This change is permanent until you do the following command /script UIErrorsFrame:Show() - re-enables the error frame, basically reverse of above. /script UIErrorsFrame:Clear() - cleans out any currently displayed error messages without blocking further ones. You can add this tithe ends of macros that you may use that can pop errors if you use them at certain times (like having the same button for 1 and 2 minute cooldowns and your 2 minute isn't up yet).