The Gemara says that the recitation of the pesukim of the korbonos is considered by HaShem as if we have actually brought them. Magen Avraham thus rules that one should stand while saying korbonos, to emulate those involved in the offering of the korbonos, who were required to stand. Shulchan Aruch HoRav says that that logic applies to mizmor l’Todah (pronounced though mizmor l’sodah) as well – one should stand while saying it. In a subsequent edition, SA HoRav says that since one is not actually a Kohen doing the avodah, it is not necessary to stand. The Ketzos HaShulchan (a sefer written on the SA HoRav) says that this applies as well to mizmor l’Todah.
It is common practice, at least among Ashkenaz daveners, to stand.
Aruch haShulchan says that a Kohen should stand—and non-Kohanim need not stand!
The late Lubavitcher Rebbe says that one need not stand during korbonos according to the SA HoRav, but for mizmor l’Todah one should stand (due to the chashivus of that perek in Tehillim—similar to that which the Shulchan Aruch writes that one should actually sing mizmor l’Todah).