I can't stand this. If Chesterton thinks he has all the power, why doesn't the onus to know "the use of it" fall on him? He should be responsible for documenting the purpose and motivations behind the fence, if he thinks he's such a protector.
Anyway, this stance seems to be what separates conservative from a liberal. Conservative want to protect something that exists but he has no knowledge of reason - such as people protecting slavery, racism, etc. Putting up barriers against people trying to change world for the better.
1. Wanting to remove the fence is a statement of "this fence does more harm than good" and if you say "there never was any reason for this fence to be built" it's very strong evidence that you did not spend sufficient time investigating the fence's benefits. Chesterton doesn't care about the protecting the fence, he cares that the person who wants to remove the fence thinks it through all the way first.
2. It's not hard to come up with reasons for slavery and racism to exist; that's all you need to do to satisfy Chesterton. I'll even go a step further and say that for societal issues if you don't have at least a basic understanding of the forces that caused them to exist (and cause them to continue to exist), then you will probably fail to change them anyways.
3. Fences take time to build. It's very rare that you get eccentric people putting fences up randomly (rare enough that discovering that to be the cause of a fence should be pretty easy because it would be so notable!)
4. Because of all of the above, I'm going to say that it's not really the difference between a conservative and a liberal, but it is part of the reason why younger people tend to be more liberal and older people tend to be more conservative. Understanding the lesson of unintended consequences puts pressure on the "conservative" side of the scale, and it can be an excuse for conservatism. However, Chesterton's fence is an important engineering principle that should be taken into account when making policy regardless of your political leanings.
Anyway, this stance seems to be what separates conservative from a liberal. Conservative want to protect something that exists but he has no knowledge of reason - such as people protecting slavery, racism, etc. Putting up barriers against people trying to change world for the better.