1) It is not clear that there is a Torah prohibition against burning pure Terumah. The Gemara (Pesachim 13a) states that on Erev Pesach, if one has pure Terumah Chametz, one must burn it in order not to transgress the prohibition of possessing Chametz on Pesach. Tosfos there (DH v'Sorfin) writes, "Even if we should say that it is forbidden to burn pure Terumah because of the verse 'Mishmeres Terumosai' (Bamidbar 18:8), in the same way as it is forbidden to cause Terumah to become Tamei..., nevertheless since the Chametz is anyway going to become forbidden on Pesach, one may burn it." One sees from the first words of Tosfos that he is in doubt about whether there is an Isur d'Oraisa of burning Terumah Tehorah. It is possible that the Torah prohibition is only on making Terumah Tamei, not on burning it.

2) It seems that the reason why Tosfos is in doubt on this matter (and uses the words, "even if we say it is forbidden to burn...") is because there is a Rishon who maintains that mid'Oraisa one may burn pure Terumah. This is Rabeinu Efraim, cited by Tosfos ha'Rashba in Pesachim there, who learns that this is indeed the reason why one is allowed to burn the Chametz there, and in fact it would be permitted mid'Oraisa even if it was not Erev Pesach.

3) According to Rabeinu Efraim we can understand how Chazal can decree that you have to burn Terumah if you touch it with Stam Yadayim, since Chazal are not overriding any Torah prohibition by this. However, many Rishonim disagree with Rabeinu Efraim, so we do not yet have an answer to your question according to the other Shitos.

4) It seems that a key source to help us understand this Inyan is the Gemara in Bechoros 34a which cites the verse "Mishmeres Terumosai" and states that we learn from this "Avid Lah Shimor" -- "make a guard for the Terumah."

5) The Gemara continues and tells us that Rebbi Yehoshua maintains "ha'Re'uyah Lecha Shemor, v'she'Einah Re'uyah Lecha Lo Tishmor" -- "if the Terumah is fit for you, then guard it, but if the Terumah is not fit for you, do not guard it."

6) We learn from this that there is no Mitzvah to protect Terumah which is not fit for consumption. The Maharit Algazi (Hilchos Bechoros, printed at the end of the Gemara Bechoros, chapter 5, #42, DH Shuv Ra'isi) writes that mid'Oraisa it is permitted to burn even Terumah which is Tamei only according to Rabbinical law, in order that people should not come to "Takalah."

7) The idea of "Takalah" is mentioned in the Gemara later in Shabbos 17b, which says that a Kohen should not keep Terumah Teme'ah in his possession lest this become a stumbling block. It must be burned in order to prevent the Terumah from being inadvertently consumed.

8) This is the Heter for the decree of Chazal to burn Terumah touched by Stam Yadayim. Chazal made a Gezeirah about touching Terumah because it is a disgrace for Terumah to be touched by filthy hands, as Rashi explains above (14a, DH Askaniyos). Once the Gezeirah had been passed, one may not eat the Terumah and therefore it must be burned in order to protect people from a stumbling block lest they mistakenly eat it.

9) We saw above that according to Rebbi Yehoshua in Bechoros 34a it can be understood why one may burn Terumah touched by Stam Yadayim. However, Rebbi Eliezer disagrees with Rebbi Yehoshua there and maintains that even "Terumah Teluyah" -- Terumah about which there is a Safek whether or not it became Tamei -- must also be guarded because of the Mitzvah of "Mishmeres Terumosai," because even though it cannot be eaten at the moment, "maybe Eliyahu will come and make it Tahor."

10) So the question now arises: According to Rebbi Eliezer, why is it permitted to burn the Terumah which was touched by Stam Yadayim? Why do we not say that since Eliyahu may come and declare that the hands of the person who touched it were not Tamei, or were not dirty, that this should require us to guard Terumah touched by Stam Yadayim, or alternatively Eliyahu may come and annul the decree of Stam Yadayim that Chazal made?

11) I suggest that we can answer this question based on the Gemara in Avodah Zarah 36a, which tells us that even if Eliyahu and his Beis Din will come and try to annul the 18 matters mentioned in the first chapter of Shabbos, they will not be able to do so. Tosfos (end of Avodah Zarah 36a) writes that this is because these Gezeiros were passed with Mesirus Nefesh, as the Gemara in Shabbos 17a tells us that "a sword was pierced in the Beis ha'Midrash."

12) Since Eliyahu can never come and change the Din that Terumah that was touched by Stam Yadayim is Tamei, this means that it is a Gezeirah that must be followed even if one knows that the Terumah is actually Tahor. Therefore, even Rebbi Eliezer will agree that it is not Terumah that can ever become permitted to eat, so he will agree that one may burn it.

We should hear Besoros Tovos!

Dovid Bloom