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?