Do electrolytic capacitors have a limited shelf life? I would like to know for both aluminium and tantalum.
Aluminium Electrolytic Capacitors:
Epcos: 2 years, cf. this applications information
Cornell Dubilier: 3 years as per this document
Nichicon: 2 years; section 2-6 in this document
Several documents say that longer storage is well possible, but will require reforming before use. Panasonic, amongst others, has a number: Apply the rated voltage via a series resistor of 1 kOhm for 30 minutes (for example https://eu.industrial.panasonic.com/sites/default/pidseu/files/downloads/files/id_almiec_e.pdf#page=186). There is also a military handbook about reforming stored electrolytic capacitors (formerly known as MIL-STD-1131).
Without reforming and by applying the rated voltage after a long storage duration, the reforming current might be so high that capacitors may get (too) warm and even blow up, which we do not like because we are not Beavis or Butt-Head (he he).
I couldn't find similar data after my initial search, but it seems like the usual MSL (moisture sensitivity levels) ratings for surface-mount parts are given and applicable.
With electrolytic caps you should always pass them through LCR meter for C & ESR check (unless you are using new caps from top-end supplier).
As long as you see that C & ESR match your requirement with some margin you can use this cap no matter how old is it. This way it's safe so reuse used caps too. Just validate them all.
Define limited: weeks, months, centuries?
For most applications the answer would be no, as long as they have been stored in conditions within spec. If the capacitors have been in hot, or very cold regions for extended time, then the electrolyte might leak out under pressure, or dry out with time. There are electronic devices that are decades old and still working just fine, capacitors and all. Sitting unused is essentially the same behavior as "shelf life."
Having said that, there is some behavior with electrolytics where the plates are "formed" when power is first applied and after sitting idle for years may need to be re-formed again. This really hasn't been an issue with capacitors for decades, so unless you're trying to use a bunch of caps that have been sitting on a shelf since 1945, I wouldn't worry about it.
I have packages of various components that are approaching 20 years old and I don't think twice before using them in new circuits.
I have got away with electrolytic caps that have been stored for up to 20 years. The storage would have been at room temp . Only reputable brands have been saved and only large power type caps have been saved for cost reasons , by large I mean thousands of microfarads . I never saved small eltec caps because they are cheap and they are supposed to dry out fast and I avoid electros in all designs for reliability reasons . The large saved electros ran fine in prototypes for audio power and energy storage . In my experience its ripple current that kills them not storage .