Happiness is calculated per building, not per person. People who lose their homes are no longer considered your citizens, so their opinion is irrelevant. Throwing a few hundred families onto the street will not affect your approval rating at all (at least not directly).
The biggest penalty you get for bulldozing buildings is that you lose their current upgrade progress. The replacement buildings will start back at level 1, which means that for quite some time they will house less people than they used to. It can then take a while until they are back to the level they had.
When you demolish and reassign a considerable percentage of your residential areas at once, you might
- suffer a population drop which might result in not just a small dive in your tax income but also in
- a shortage of workers which leads to
- a lot of offices, industry buildings and commercial buildings shutting down (also resetting them to level 1 when reclaimed) which means that
- your tax income takes a very large dive
This means that large-scale replanning of your city layout should not be done all at once but step by step with breaks in between to let the newly assigned quarters develop.