In Pesachim 23a, the gemara (in the name of Rav Papa) derives from Numbers 18:27 that the prohibition of terumah to non-priests only applies to eating and drinking, but not to other forms of enjoyment. The example given in the gemara pertains to the mishna in Eruvin 3:1, to the effect that a yisrael can use terumah for his eruv.
According to this article at Virtual Beit Midrash, this statement in the gemara has been understood to mean that terumah is only forbidden for consumption by yisraelim, but not for regular enjoyment (מותרת בהנאה לזרים). As the author of the article (ר' מתן גלידאי) makes clear, this ruling created problems for the rishonim, concerning how to align it with the mishna in Terumot 11:9. There, the mishna makes clear that kohanim may feed their animals terumah, but that yisraelim cannot. If there is no prohibition of deriving enjoyment from terumah, why would a yisrael be prevented from feeding it to his livestock?
The resolution acc. to that same article, which was proposed by Rabbeinu Tam, is to the effect that the Torah permits yisraelim to enjoy terumah, but forbids them from any form of enjoyment that results in the eradication of that terumah (הנאה של כילוי). This explains why a yisrael would be forbidden from feeding his livestock with terumah, as well as why the mishna in Terumot 11:10 would permit his burning terumah for the public good - as long as it was with the permission of a kohen.
But how does it align with the mishna in Terumot 10:4? There, the mishna speaks of somebody lighting his stove with cumin of terumah (תנור שהסיקו בכמון של תרומה). The discussion as to whether or not bread cooked upon this stove is permissible for consumption is one that pertains to the separate issue of whether or not aroma is a tangible substance (ריחא מילתא היא), but so far as the actual burning of the cumin is concerned, the following is what the Bartenura has to say:
ואי משום דהוסק התנור בתרומה, אין איסורה איסור הנאה
And concerning the fact that
he lit the stovethe stove was lit with terumah, its prohibition is not a prohibition of enjoyment [but only of consumption].
How does this align with the prohibition of הנאה של כילוי, as expressed by Rabbeinu Tam?
To compound this problem, both opinions are brought (contradictorily) in Kehati's peirush on the Mishna. In his commentary on Terumot 10:4, he quotes the Bartenura as follows:
ומה שהוסק התנור בתרומה אין בזה איסור, שאין התרומה אסורה לזרים בהנאה אלא באכילה בלבד
And concerning the fact that
he lit the stovethe stove was lit with terumah, this is not prohibited, since terumah is not prohibited to non-priests when it comes to enjoyment but only when it comes to eating [and drinking].
Later, however, in his introduction to Terumot 11:10, he explicitly declares the following:
ואף על פי שתרומה אינה מאיסורי הנאה, ומותרת היא בהנאה לזרים, מכל מקום הנאה של כילוי, כגון בענייננו שההנאה באה משריפת התרומה, אסור היא לזרים
Even though terumah is not of those things that are forbidden from enjoyment, and is permitted for the enjoyment of non-priests, the enjoyment that comes from eradication (הנאה של כילוי) - such as is found in this instance, in which the enjoyment comes from the burning of terumah - is forbidden to non-priests.
In summary, the two ways of putting this question are as follows:
If yisraelim are allowed to enjoy terumah so long as they do not destroy it, why does the Bartenura suggest that burning it (in Terumot 10:4) is okay?
How can Kehati explicitly say that burning terumah (in Terumot 10:4) is okay, but then say no less explicitly (in Terumot 11:10) that burning terumah is impermissible?