What kind of damage could a soldering iron do to a component?
My experience is that there are several categories of things that can happen. I think it's easiest to group them by component type. I learned these all the hard way. Note that several of these involve the use of heat and force at the same time. In general this is not a good idea. Most parts can tolerate far more of either, alone, than they can of both combined.

Low temperature plastic components

This includes encapsulated DC/DC converters, connectors, switch bodies, etc. These can and will melt with sometimes terrifying ease. The good news is that most of the time the damage is cosmetic. The bad news is that if you care about how the board looks, well...

Also you usually can't tell up front what will melt and what won't, without an 'experiment'. 

Sometimes it's smarter to pull vulnerable parts off the board and then reinstall them later.

Leaded encapsulated components

Through hole, encapsulated components with through hole leads (DC/DC converters or transformers). Too much heat combined with pulling and the lead comes neatly out. If you are lucky, the lead will fall out or be pulled out during rework. Otherwise it's a debugging problem.

Insulated wire

IDC (ribbon) cable is notorious for this. The slang term is "marshmallowing". If you've ever caught a marshmallow on fire, you know why. The insulation melts, burns, bubbles, etc. This takes skill to avoid, especially with softer insulation.

Of course bumping a wire bundle with the barrel of the iron, after everything is soldered in place, is a great trick too.

Printed Circuit Boards

I include these for two reasons. First, many breakout boards are used themselves as components. Second, the main PCB is itself an important component in the design.

The big things with circuit boards are burning, lifted traces, or gouged solder mask. Burns happen when the iron is too hot. Unmasked boards seem more vulnerable than solder masked boards. Loosened traces/pads and solder mask damage happen when you apply too much force with the soldering iron (trying to get that stubborn lead loose).

Warping of the PCB is possible but you have to try hard. Thin PCB + excess pressure + dwell time = permanent curve.

Integrated Circuits

I have never (yet) killed an IC with a soldering iron. I have damaged and destroyed SMT chips with hot air rework tools though (that's another topic). Most chips have a maximum lead temperature/time rating, so I think it's possible.

Passive SMDs

These go bad when you are trying to install them, and they stick to the iron. While you're busy trying to get them loose and not lose them, they can cook. Usually one of the terminals comes loose, and usually that happens when the part's half soldered on the board. You can also cook chip resistors this way until they visibly discolor - IMO, that's a throwaway. Of course the smaller they are, the less mass they have and the easier it is to do this. 0201 resistors for instance, take some getting used too (buy many spares).