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).

In my experience it is not always obvious that damage has occurred. It isn't until I plug it in and stuff is acting odd that I know I did something bad.

I have actually found that passive components seem to take a bigger/quicker hit then ICs do. I have seen it mostly with resistors changing their resistance or capacitors acting as shorts and/or opens.

If I were to guess, overheating a component would probably lower the life span of the component. ICs just aren't designed to go through very much thermal fatigue. This probably isn't an issue if you are just using the device for a short duration, but could be very bad if it was a fix for a customer.

You might not notice anything immediately wrong, but running a component up to molten solder temperatures is likely to short the device's life span. It only takes a couple of seconds to get serious amounts of heat into a part. (Try it with a disc ceramic capacitor some time; by the count of three, you'll see the surface of the cap go all shiny and wet looking as the coating melts!)

As for protecting parts, this doesn't help for IC's or very small components, but if you want to protect discrete components that have wire leads, it's very easy to clip a small alligator clip on a lead, between the device and the solder point, to sink heat away from the device. This works well for small capacitors, low-wattage resistors, even TO-92 and TO-220 cased semiconductors.

Along the same lines as alligator clips, you can use 'medical forceps' if you have those, or even a small pair of needle-nosed pliers with a rubber band wrapped to keep the jaws shut.