Liberty: Being free of constraints
Efficiency: Raising the state of the world as high as possible on everyone's preference ranking (or some aggregate measure thereof)
You might have heard of Amartya Sen's Liberal paradox, which purports to show that the two necessarily conflict. Of course, as I said a while back, it does no such thing; it only shows a problem with preventing people from waiving their liberties when they find it preferable to do so.
However, there is a real sense in which those two conflict, and it becomes most apparent in discussions of taxation, and how to make it better.
The conventional economist's view is that "The ideal tax system is the one that hurts efficiency the least."
But there's another view, exemplified by the Murphy article that I linked in my last post: "The ideal tax system is the one that's easiest to opt out of."
Naturally, these really do conflict. Why? Because generally speaking, if you want to levy a tax that merely transfers purchasing power to the government without also forcing people to bear other hardships, you have to do it by taxing goods with inelastic demand, like energy, as people will not respond to the tax by buying less of the good, which would indicate a reduction in efficiency.
But the harder a tax is to avoid, the harder it is to "opt-out" of!
So if you think it's good for people to be able to legally reduce government revenues by abstaining from a product at relatively little cost to themselves, then "economically efficient taxes" are no longer an unvarnished good, as they come at the direct expense of the goal of making it easier for people to change their behavior in a way that routes around taxation.
This, I think, is the true conflict between efficiency and liberty, as it doesn't hinge on confusing rights and obligations.