I've been told that the language of the libretti penned by Piave (for Verdi) and Illica (for Puccini) are written in some kind of "old" and "dated" language that no Italian of today can easily understand without training. Is this true?
No that is completely false. While the language is clearly poetic (and so a little "harder" than everyday conversation) and from 150 to 100 years old, it is not substantially different from standard Italian and can be understood by any native speaker.
As a proof I bring myself at the age of 13, when I first went to La Traviata and L'Aida without having any problem.
This in my experience is valid for all opera librettos at least from Pergolesi onwards (Baroque opera is a slightly different matter because Baroque poets loved playing with the language a bit too much; it is still understandable with a mild effort though).
Moreover a lot of expressions from the librettos of di Piave actually migrated into sayings, somehow similar to what happened to English with Shakespeare. For example croce e delizia (to mean that something is both a pleasure and a sorrow) and i bollenti spiriti (to mean the impulsiveness, especially of youth), both from the libretto of La Traviata.
EDIT: To answer the criticism in the comments. Of course the language is a little harder than usual spoken Italian: it is poetry. However I think you are underestimating the capability of the average person. In my experience everyone can understand those words and I think it is very dangerous to perpetuate the idea that they need some special training to do so. This myth is in fact part of what pushes people away from even trying. Ok I'll stop here or it becomes a rant; so if you still disagree, downvote away :).