The content of the śloka is not to be taken literally in this sense. The translation provided is a bit weird (since it uses the term soul instead of Self, and faith which isn't there in Sanskrit śloka) and can cause much misinterpretation of the original text.(I guess it's an ISKON translation)
The killing of the Self doesn't mean killing of the Self literally but being ignorant of the Self is shown here to be equivalent to killing it; just as imputing a false and serious charge against a virtuous man is, in common parlance is considered to be 'murder without weapon'—aśastravadha. On account of the ignorance of the Self, one also is ignorant of the nature of the Self and its knowledge, thus being subject to birth and death.
asuryā nāma te lokā andhena tamasāvr̥tāḥ |
tāṁste pretyābhigacchanti ye ke cātmahano janāḥ ||3||
Trans. (by Gambhirananda) : Those worlds of asuras are covered by blinding darkness. Those people that kill the Self go to them after giving up this body.
Śaṅkara's bhāṣya on the above śloka explains the content quite lucidly, which is as below
asuryāḥ, of asuras; as compared with the attainment of non-dual state of the supreme Self, even devas are asuras and the worlds belonging to them are asuryāḥ. The word nāma is a meaningless indeclinable. Te, those ; lokāḥ (lit. worlds)— (derived) from the root luk —means the births in which the results of karma are perceived or enjoyed ; āvr̥tāḥ, are covered ; andhena, by blinding —characterized by the inability to see ; tamasā, by darkness—in the form of ignorance. Tān, to them—that extend up to the motionless (trees etc.) ; pretya, after departing, giving up this body ; gacchanti, go—in accordance with their karma and meditation (on devas etc.) ; ātmahanaḥ, those that kill the Self. Who are they? Janāḥ, (the common people) those that are ignorant. How do they kill the eternal Self? Because the Self, which exists, is concealed through the fault of ignorance. The experience of the Self as free from decrepitude and death (present in the realization, "I am free from decrepitude and death"), that comes as a result of the existence of the Self, remains concealed, as is the consciousness of a person who is killed. So, the ordinary and ignorant people are called the killers of the Self. Because of this fault of slaying the Self, they are subject to birth and death.
The commentary explains it quite smoothly. But ignorance being referred to as killing of the Self is also referred in the Taittirīya Upaniṣad (2.6.1)
asanneva sa bhavati | asadbrahmoti veda cet |
asti brahmoti cedveda || santamenaṁ tato viduriti |
Trans. (Gambhirananda) : If anyone knows Brahman as non-existing, he himself becomes non-existent. If anyone knows that Brahman does exist, then they consider him as existing by virtue of that (knowledge).
In Bhagavad Gītā (13.28), we see that by knowing the Self by seeing it everywhere one doesn't annihilate the Self by the Self, which quite clearly connects with Īśopaniṣad (verse 3)
samaṁ paśyanhi sarvatra samavasthitamīśvaram |
na hinstyātmnātmānaṁ tato yāti parāṁ gatim || 28 ||
Trans. : Since by seeing God who is the same as uniformly present everywhere he does not annihilate the Self by the Self, therefore he reaches the supreme Goal.
In Mahābhārata, Śāntī Parva (1.74.27) we see a reference of ignorance regarding the Self being considered as a sin, "What sin remains uncomitted by that thief who steals away the Self, who comprehends the Self other than what It (really) is! " The aforementioned reference from Mahābhārata is also mentioned by Madhusūdana Sarasvatī in his Gūḍhārtha-Dīpikā on Bhagavad Gītā (13.28).