Parashat Tzav contains a passage that details the laws of eating the qorban todah.

This is the ritual of the sacrifice of well-being (shelamim) that one may offer to the L-RD. If he offers it for thanksgiving, he shall offer together with the sacrifice of thanksgiving unleavened cakes with oil mixed in, unleavened wafers spread with oil, and cakes of choice flour with oil mixed in, well soaked. This offering, with cakes of leavened bread added, he shall offer along with his thanksgiving sacrifice of well-being. Out of this he shall offer one of each kind as a gift to the L-Rd, it shall go to the priest who dashes the blood of well-being.   And the flesh of his thanksgiving sacrifice of well-being shall be eaten on the day that it is offered; none of it shall be set aside until morning (Leviticus 7:11-15).

The last law, that in contradistinction to other shelamim, qorban todah must be eaten only on the day that it is offered (and the following night), but not on the following day as well, reminds us of the laws of qorban Pesah.

You shall not leave any of it over until morning; whatever is left of it until morning you shall burn (Exodus 12:10).

I think that it is appealing to argue further that qorban Pesah is itself, from a philosophical perspective, a type of qorban todah. Yetzi’at Mitzrayim was certainly the quintessential act of beneficence to the Israelites, worthy of a specific reciprocal gesture in return. That notion is mirrored in every qorban todah, when one is delivered from jail, the perils of a sea voyage, a journey through the desert, or a sickness (Psalm 107, which is interpreted in the halakhah as the motivating factors for the bringing of a qorban todah). The halakhic fact that in both cases, one may not eat of the qorban on the following day reinforces the conceptual similarity of the two sacrifices.

Ramban in Parashat Tzav suggests that ideally, all qorbanot shelamim must be eaten on the day that they were brought (and the following night.) The permissibility of eating the “leftovers” on the second day does not express the le-chatkhila norm. He writes as follows:

In light of the words of the Ramban, one may suggest that although in general, God allows us to eat the meat of the shelamim on the second day after it was brought, in certain situations that permission cannot be given. For when one eats leftovers on the second day, one is primarily performing an act of self-gratification (eating), and the religious impulses that accompanied the initial bringing of the sacrifice have already begun to fade from one’s consciousness. Eating meat on the second day is more fundamentally a secular act than eating on the first day. With regard to qorban Pesah and qorban Todah, on the other hand, one’s consciousness of gratitude towards God must be constantly paramount. Hence, the time for eating is restricted to one day and the following night.