In general, you get the least trouble when the circumference of the new tire is close to the old tire. So when you move to bigger rims, you also need lower-section tires.
You can use a tire size calculator to find a combination that works well.
When the circumference of the new tire is much larger than the old one, your speed will be higher at the same rpm, your speedometer reading will be incorrect (too low), and acceleration will suffer.
You can also get problems with the wheel hitting the bodywork.
When you go to a lower-section tire, the ride gets a bit less comfortable (the tire can't absorb bumps as well because it's stiffer), but handling improves (a stiffer tire means it allows less sideways movement of the rim relative to the contact patch).
Bigger rims can also increase unsprung weight, which may mean the ride over short bumps gets less comfortable.
This could stress the shock absorbers a bit more, but I don't think this is significant. They're rated for the maximum weight of the car, which is a lot more than the few kg you can add in unsprung weight.
Unsprung weight doesn't change the ride height. The tire diameter can change the ride height, but you need to go to a much bigger tire size to make a visible difference in ride height.