This question pertains to a related question.
Leviticus 7:20-21 says:
כ וְהַנֶּפֶשׁ אֲשֶׁר-תֹּאכַל בָּשָׂר, מִזֶּבַח הַשְּׁלָמִים אֲשֶׁר לַיהוָה, וְטֻמְאָתוֹ, עָלָיו--וְנִכְרְתָה הַנֶּפֶשׁ הַהִוא, מֵעַמֶּיהָ.
But the soul that eateth of the flesh of the sacrifice of peace-offerings, that pertain unto the LORD, having his uncleanness upon him, that soul shall be cut off from his people.
כא וְנֶפֶשׁ כִּי-תִגַּע בְּכָל-טָמֵא, בְּטֻמְאַת אָדָם אוֹ בִּבְהֵמָה טְמֵאָה אוֹ בְּכָל-שֶׁקֶץ טָמֵא, וְאָכַל מִבְּשַׂר-זֶבַח הַשְּׁלָמִים, אֲשֶׁר לַיהוָה--וְנִכְרְתָה הַנֶּפֶשׁ הַהִוא, מֵעַמֶּיהָ.
And when any one shall touch any unclean thing, whether it be the uncleanness of man, or an unclean beast, or any unclean detestable thing, and eat of the flesh of the sacrifice of peace-offerings, which pertain unto the LORD, that soul shall be cut off from his people.
According to a relevant article on myjewishlearning.com:
Rashi, an 11th century Torah commentator, explained that this prohibition against touching pigs only applied when Jews were on their way to Jerusalem to observe the three pilgrimage festivals–Passover, Shavuot, and Sukkot. During those festivals the people were required to be in a heightened state of purity, so they had to avoid touching something like the flesh of a pig. Even on the way to Jerusalem, Jews were only prohibited from touching the flesh, that is, the meat of a pig.
Rashi considers the context to discern that it is being unclean while eating the peace offering that would cause one to be cut off from their people. This, however, does not change the fact that the result of touching an unclean animal is that one becomes unclean.
This makes me wonder how organ transplants from unclean animals will affect people going up to Jerusalem when the temple is rebuilt. Will they die?
One comment points out that touching a dead body also renders one unclean so the same questions might be asked in regard to transplants from human cadavers.