Physics principle: Angular Momentum is conserved in a closed system.
The only way you are going to get a momentum change is by:
- transferring momentum to/from another body in the system.
- shifting momentum closer/further away from the center of rotation.
- shifting the center of rotation.
Your not really going to have much luck here in a universe as we know it sense, by human hands. Our best achievement to date is the Three Gorges Dam which has shifted the mass of the earth out slightly enough to increase the length of a day by about 0.06 microseconds.
If you were to build an even more massive and taller structure slightly off balance from a pole, you might get the Earth to very slowly increase its natural wobble. Currently our poles slightly wobble in a multi-millenia axial precession. Unfortunately our structures don't tend to last the hundreds of thousands of years needed to have a meaningful impact.
Alternately, if the earth's center is not what we thought it was, have the primary mass move off center. As a significant part of earths mass is in the core this will have a non-trivial effect on rotation. How you manage to move so much mass against gravity and pressure though is beyond me.
You might be able to accelerate the wobble more effectively by looking out into space.
The most obvious candidate is the moon. Due to its size and relative proximity, it actual acts to balance out the earth's rotation, keeping it relatively smooth. Otherwise the oceans, and the air on earth could cause day lengths to vary by hours. So go out there and invest a lot of energy in changing the moons elevation, and the inclination of its orbit. Earths rotation and tilt will suffer. Unfortunately while humans may have the technology, they do not have the industry or political will to actually maintain the effort needed for this right now (or even within the next 100 years).
On the other hand, why make the earthlings do all the heavy lifting? Having a moon sized object pass close (moon orbit close) to earth, or a much smaller but still hefty piece of rock hit earth (or the moon) could in theory impart enough energy to change the angular momentum.
If it were to hit earth, it would need to be bigger than the asteroid that killed the dinosaurs. Any humans alive are probably in orbit, and they are contending with millions of little projectiles shot into orbit from the impact. It is possible that the earth is now mostly molten lava.
If it hit the moon, earth is fine (for now), but the space industry will not like the new environment much, and civilisation will live in much greater fear of asteroid impacts (parts of the moon).
If instead it missed, wandering off into space, it may have had enough influence to alter the momentum in the system and cause a tilt. Alternately if it hangs around it will act like another moon. The new planet and two moon system will shift angular momentum around differently making the rotation/tilt more regular/chaotic. With a new moon the tides will change. Either way Weather will be affected in the short term, perhaps even permanently.