The first thing I'd be worried about is how fast the change in axial tilt would happen.
If it happens in minutes or seconds, then people get thrown into walls at high velocities. The walls won't survive.
If it happens over the course of a day, there would be major tectonic stresses suddenly redistributed. Earthquakes, volcanoes, dogs and cats living together... Mass hysteria!
Even if it happened over the course of several months, there is still a huge mismatch in rotational energy, where the core is spinning in a different direction from the crust, causing huge stresses.
But, if it happens over the course of a day, it still allows for an interesting story. Some buildings might slip from their foundations, as the forces of inertia suddenly make gravity seem that "down" has shifted by a couple of degrees for several hours, but most buildings and technology would survive.
Astronomers are pissed off. Besides all of their telescopes pointing in the wrong direction, their favorite observatories that used to sit on some of the tallest and most remote mountains are sitting in active volcanic calderas.
People also can't get satellite TV unless they reposition their dishes.
And, somewhat more significant to the future: Jet airplanes would be grounded. Not because of any property of axial tilt, but because of volcanic ash and how destructive it is to jet engines.
The first climate effect would be the same, whether you increased or decreased axial tilt:
Volcanic ash is pretty reflective, and doesn't hold heat very well; quite the opposite to atmospheric CO2, which doesn't reflect light but holds heat very well.
Global temperatures drop quickly. Some microclimates may warm up -- i.e., decreased turbulence would keep tropical air from mixing with air in cooler regions as much, but overall the average temperature drops quickly and significantly. (See Little Ice Age for an example of just a few more volcanoes than usual being active over a long period of time.)
Ash doesn't stay in the air for long. Tectonic stress doesn't stay high for long when the stress is actively being relieved. This volcanic winter should only last for a decade or so.
It's likely to be followed by a volcanic summer. Ash on the ground gets wet, gets dark, and traps heat on the surface. Humans will have probably used much more fuel to warm their homes and try to plant more crops, releasing more CO2, which will now start trapping that heat.
But, the next effect depends more on which direction you tilted the axis. Is the tilt now 13 degrees or 33 degrees?
If the tilt were only 13 degrees, then seasons as we know it would greatly diminish. Areas at the equator would get much warmer. Areas near the poles would get much colder. There would be much less circulation between these regions. The tropics (which would be a much more narrow band around the equator) would see more rain, while the rest of the world would see much less. Much ecological diversity would be lost, and humans would have to struggle to adapt.
If the tilt were increased, then seasons would get far more pronounced. Deserts would get rain. Rainforests would dry up. Much ecological diversity would be lost, and humans would have to struggle to adapt.
Note that this answer is based on an earlier version of the question, where the change of axial tilt was limited to 10 degrees, and there were no limits on how fast the change would occur.