The syntactic difference, as lonehorseend points out, is that in each pair, one infinitive is active and the other is passive:
The past is something to forget.
XX is the last thing to mention in this study.
The past is something to be forgotten.
XX is the last thing to be mentioned in this study.
The semantic difference varies with context.
Your first pair of sentences seem to represent a philosophic recommendation, a task which you lay before everybody-in-general and nobody-in-particular. Consequently, the implicit subject of the active version is identical with the excluded subject of the passive version:
The past is something for everybody to forget.
The past is something to be forgotten by everybody.
There is no real difference between the two.
This may also be true with the second pair. For instance, if you are the author of the study in question you may use either, indifferently, to announce the final task before you, your closing topic:
XXX is the last thing for me to mention in this study.
XXX is the last thing to be mentioned by me in this study.
But for a reviewer of the study in question, the situation is quite different. The active version makes no sense in this context, because the implicit subject—the only available subject for the infinitive to mention—is the writer of the sentence: the reviewer. That would illogically make mentioning XXX in the study the reviewer's task rather than the author's.
The passive version, however, has an explicit subject, the last thing. Consequently, the clause may bear an alternative interpretation: the last thing to be mentioned may be parsed as the last thing which is mentioned. This makes perfectly good sense within the context:
XXX is the last thing mentioned in this study, and it seems to me it deserves a fuller treatment.
Context, context, context.