The syntactic difference, as lonehorseend points out, is that in each pair, one infinitive is active and the other is passive:
ACTIVE
The past is something to forget.
XX is the last thing to mention in this study.
PASSIVE
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.
"Got his" adds an ingredient of blame over the person involved, meaning that he/she contributed in some way to the event.
You can be told not to drive into a dangerous neighborhood, because you can "get your car stolen" if you do so.
Or you can "get your nose broken" if you provoke the wrong person.
On the other hand, if no negligence was shown by the person involved in the event, you can simply say that "his car was stolen" and "his nose was broken".
Best Answer
Every verb has its own rules for what sorts of complement it permits and what sorts it forbids. These rules are in the end completely arbitrary, and there's no rule or general principle you can point to say “Why” one verb works one way and another verb works another. It’s a matter of historical contingency; at some point the language drifted toward doing it that way and not another way.
You can in some cases point to possible historical influences. In this case, for instance, have has been acting as an auxiliary for a very long time, over a thousand years, and the particular sorts of complement it takes were determined before the infinitive marked with to existed in its current form. Get, however, was only pressed into service as an auxiliary much later. From the 15th into the 17th century the language experimented with get X for to VERB-inf and get VERB-inf X before settling on get X to VERB. But there's no rule there that will predict any other verb's complementation pattern; history is only useful as a mnemonic respecting get.
That’s how idioms work: you have to learn them one by one.