Fundamentally, there are two components to the belief that you shouldn't pay your children for every chore they do:
1. Children should participate in family chores because they are members of the family, not just because they are paid. Not every family member gets paid for washing dishes, doing the laundry, cleaning their rooms, or taking out the garbage. The flip side of this is that children -- not adults -- should have an allowance because one of their developmental tasks is learning how to manage money. We do those things because we're part of a social group -- a family.
2. Children should have an opportunity to earn additional money for additional work. While they should know what is expected of them as family members, they should also know that they can get greater rewards -- including extra cash when appropriate -- for doing extra work.
The reason for the first is that children (and many adults) tend to be "target earners." If your eight-year-old has enough Pokemon cards for the time being, he'll tell you that he won't take out the garbage (or whatever his assigned chore is) because he doesn't need the money. By paying for all chores, parents give the impression that you don't have to help out if you don't need the cash. Is this really the lesson that RicsAst and others want to teach?
The reason for the second is akin to what RicsAst suggests. In fact, it goes well beyond what he's written. Children take a great deal of pride in earning money. It helps them learn who they are and how they fit in -- not only within their family but within society in general. Of course, it also allows them to get that bike or guitar they've always wanted. In fact, they'll value those purchases even more if they make them with money they've earned.
So setting up "money for chores" vs. "no money for chores" is a straw man. We need to give children both the responsibilities and the opportunities, not a choice between them.