That's an interesting question. Presumably the issue may relate:
(a) to the material from which ping-pong balls are made, and
(b) from the adhesive used to attach them.
(c) local effects on the growth and health of the underlying nail.
(a) Traditionally ping-pong balls were made from celluloid which is non-toxic if it is in good condition (though, worth noting, it is very inflammable). The manufacture of celluloid may be hazardous, because of the chemicals involved, and old celluloid can break down into nitrocellulose and camphor - with further breakdown producing unpleasant chemicals such as nitric acid. Nitrocellulose, by the way, is an explosive - so maybe if you use old, brittle cellulose you may get exploding nails
I believe ping-pong balls are increasingly made of more modern, stable plastics which presumably are no hazard and, to be serious, I doubt whether you'd ever get problems with recently made celluloid, but it would be a mistake to stockpile old celluloid for this purpose.
(b) Probably more pertinent are the possible toxic effects of adhesive. As far as I know the adhesives used for artificial nails are all cyanoacrylate glues (termed 'superglue' in the UK). These are generally regarded as non-toxic when set. But the fumes of the raw glue can be irritant and inhaling them can lead to an allergic response. I believe there is one form of cyanoacrylate that has been developed for use in medical applications (in particular as an alternative to suture in wound closure). This is 2-octyl cyanoacrylate; the risk of allergy seems to be much less with this. A bit of googling will show you that this adhesive is available over the counter in a proprietary formulation.
(c) I'm not sure of the local effects of long term artificial nail use but there are plenty of warnings around this - mainly based on issues from nail salons. My impression from using cyanoacrylate and silk to mend broken nails is that occluding the natural nail in this way does have a negative effect and I think this is the issue that would keep me away from anything other than short term use of artificial nails.
Disclaimer: I am not an expert in this matter but in view of my background in both chemistry and medicine I maintain a general interest in issues of this sort. And to answer the question, I myself would not use artificial nails long term.