Is it worth reusing old components
I guess the answer is maybe.
Yes, if...
you're in the process of fixing something and the hard-to-get spare part is readily available on a junk board in the corner of your lab. Hard to get can even be a standard 10k resistor, if it's Sunday afternoon and you don't want to order parts online or wait for the next electronics shop to open. Sometimes, I even re-use old parts when hacking something at my company and don't want to wait two days for ordered parts before putting the design in production.
you're trying to learn from an existing design and rebuild a sub-circuit found on a larger board. Besides solid knowledge about the required theory, existing designs are your best teacher.
you're after exotic components (high voltage from CRT monitors, vintage parts from old equipment, cool-looking stuff like nixie tubes or a magic eye from an old radio)
No, if...
you're building a large new project and the components we're talking about are standard, cheap diodes, resistors, ...
you're going to sell your circuit and must know where your components come from. Who tells you that a part on a junk board has never been subjected to stresses beyond those listed in the data sheet's absolute maximum ratings, causing permanent damage or deviations from the values you must count on?
For me, the decision depends on a few criteria:
- Speed
- Reliability
- Cost, availability
When to use a junk part
If I can't wait for the part and don't have on in my materials, I'll turn to the junk pile.
If I can verify the part will operate correctly within parameters (ie, isn't damaged), or the project doesn't require high reliability, I'll turn to the junk pile.
If the part is expensive, rare, or has some intrinsic value that a new part won't come with, then I'll turn to the junk pile.
When to use a new part
If I have the part new in stock, I'll generally avoid the junk pile because reuse takes longer than using a new part.
If I cannot easily verify correct operation of the junk part without a lot of effort, or the project/part is going to a client or will be used by others than myself, I'll typically wait to get a new part.
I'll tend towards using the new part by default because even if the part is good, when I run into a problem I tend to assume it's a problem with re-used parts of unknown reliability. This causes me to spend more time debugging issues than I would with new parts.
Sometimes when I just have a spare evening and want to get my mind of off more important matters, I take a random PCB from my "junk bag" collected from some strange contraptions, and solder out everything that looks cool from it. That's how I got my hands on lots of totally rad, but also 95% totally useless ICs, some mysterious parts which I wasn't able to identify despite many sleepless nights spent digging through photos and datasheets, and some totally regular and common stuff, which was probably worth about 1/10th of the juice my soldering iron used in the process. Still, soldering is a nice way to heat up in winter, and the smell of fresh solder in the morning is something a true man will surely appreciate.
OTOH, a) you can get brand new samples for free in stock amounts from many vendors (YMMV), b) most of the stuff you'll get from disassembly will probably be worthless and/or neither reprogrammable nor refittable for any serious purpose, and you'll never know when they'll fail direly, requiring you to fix entire circuit that got reckd, c) some parts will get FUBAR during your "board games"... still, you'll probably learn how to solder efficiently after about 100 hours spent on desoldering/soldering them.
As a result, I have a nice box with lots of brand-new samples & fresh-from-the-shelf stuff, which I use for brand-new fresh-from-the-shelf ideas, and a bag of old junk, which I use for all those "it'll probably burn or explode... SO LET'S DO IT NOW!" ideas. I think (no citation nor hard proof here, sorry) that most people with some hobbyistic EE experience have a system similar to mine.
tl;dr the only real answer is, quoting zebonaut - maybe. If you're doing EE for money - don't bother with reuse even for prototypes or tests; it's essentially like asking for trouble. If you're doing it for fun though, the rule is more trouble == more fun instead - so the answer is "Of course it's worth it, it still probably has the magic smoke inside!".