The Pesach offering is slaughtered for the sake of being eaten, since that is its own particular mitzvah - it is designated to be eaten. Therefore it must be slaughtered for the sake of people who are able to eat from it, and if it was slaughtered for the sake of those who are not fit to eat it, it is disqualified.
Slaughtered it for those who cannot eat it - liable:
The Mishnah lists those categories for whose sake, if the offering was slaughtered, the Pesach is not valid:
"shelo l'ochlav" - people who are unable to eat from it, such as the sick or the very elderly, who do not have the strength to eat from the Pesach.
"shelo limnuyav" - people who were not registered in advance for this particular Pesach offering. One must register and join a group for each and every offering, and a person may not eat from someone else's Pesach offering.
"la'areilim" - those who were never circumcised, who are not fit to eat from the Pesach.
"v'litmei'im" - those who are tamei, who likewise are not fit to eat from the Pesach.
One who slaughters for their sake on Shabbos is liable to bring a sin offering, since this is not a valid Pesach offering, and it turns out that he desecrated Shabbos for no purpose.
Slaughtered it for those who can eat it and for those who cannot - exempt:
What is the law when the slaughter was done for a mixed group: "l'ochlav v'shelo l'ochlav, limnuyav v'shelo limnuyav, l'mulim v'la'areilim, l'tehorim v'litmei'im"? In such a case he is exempt, because the offering is valid. As long as those included in the slaughter are people who are fit to eat from it, the fact that he also slaughtered for the sake of those who are not fit does not disqualify the offering, and therefore even if he slaughtered on Shabbos there is no desecration of Shabbos here.
Slaughtered it and it was found to be blemished:
If he slaughtered on Shabbos and it became clear that the animal was blemished and not fit for a Pesach offering, he is liable to bring a sin offering. The Pesach is disqualified, so it turns out that he desecrated Shabbos, and the one who slaughtered is not considered to have acted under circumstances beyond his control but rather unintentionally, since it was in his power to examine the animal beforehand. Since he did not examine it, there is a measure of negligence here, and therefore he brings a sin offering.
Slaughtered it and it was found to have a hidden fatal defect:
If he slaughtered and it became clear that the animal was a tereifah - such as an adhesion on the lung or another defect in the internal organs - he is exempt. Here the one who slaughtered acted under circumstances beyond his control, since the defect is hidden and the matter is entirely outside his control, and there is not even unintentional negligence here. The offering is indeed invalid, but the one who slaughtered is not liable to bring a sin offering for desecrating Shabbos.
It became known that the owners withdrew, that they died, or that they became tamei:
He slaughtered the Pesach, and afterward one of the following became clear to him:
"shemashchu ha'be'alim es yadam" - those very people for whose sake he slaughtered had withdrawn from this Pesach offering and registered for a different one.
"o shemeisu" - the owners had died in the meantime.
"o shenitme'u" - they became tamei, and can no longer eat from this offering but are deferred to Pesach Sheni.
In these cases too he is exempt from the sin offering, because the one who slaughtered acted under circumstances beyond his control: this is a mistake that he had no way of knowing about or controlling, and it is not considered unintentional negligence.
"mipnei sheshachat b'reshus":
The reason for the exemption in all these cases is that at the moment of slaughter he slaughtered while he was permitted to do so. A person is not obligated to investigate and verify whether the owners are still alive, whether they are tahor, or whether they have perhaps withdrawn from the offering; he is entitled to assume that the situation remains as it was, and he is not expected to worry that things may have changed.
In summary: One who slaughters solely for the sake of those disqualified from eating is liable, since the Pesach is invalid; if he slaughtered for a mixture of those who are fit and those who are not, he is exempt, since the offering is valid; if it was found to be blemished, he is liable, since he should have examined it and is therefore considered to have acted unintentionally; if it was found to have a hidden fatal defect, or it became known that the owners withdrew, died, or became tamei, he is exempt, since he acted under circumstances beyond his control and slaughtered with permission.