A little while ago, I encountered a
nice rendition of the old folk song "Leave Her Johnny". It reminded me of my own one-verse filk:
I heard old Archimedes say,
"Lever, Johnny, lever
With a place to stand I could move the world
If I only had a lever"
When I look up the quotation on-line, the comments are mostly about the need for a lever of astronomical scale. But technically, that's not true. A lever of any size will do, and the fulcrum can be on the Earth's surface.
Any shifting of mass, even mass that's part of the Earth as a whole, will affect the planet's moment of inertia and how it rotates. It's just that when we're talking about the kinds of masses common in our lives, the changes will be tiny and undetectable. The changes are inversely proportional to the ratio between the mass of the stuff being shifted and the mass of the rest of the planet.
Though seismological stations have to be set up in such a way that vibrations from local events are cancelled out. That stuff also moves the world... a little.