You need to examine what the "obstacles" are doing in this sentence.
The obstacles prevent people who do business from doing it.
So it is an obsticale to doing bussiness.
The people who do business can not do business with these obsticales in place. They are prevented from doing business.
It is the people who are doing the business not the obsticale.
Does that help? Or is it more confusing?
Also I have corrected your post.
Could anyone please explain to me the structure of the following sentence?
"the government is doing nothing to dismantle many of the obstacles to doing business"
Why use "to doing business" rather than "to do business"?