Ykhjuy

What is the difference between Naishtika Brahmacharya and Sanyasa. Specifically in the context of 'Agni'. Since Sanyasa involves 'agni-tyaga'. Does Naishtika Brahmacharya also involve 'Agni-tyaga'?

A Naishtika Brahmachari is perpetual student. He lives till his death the life that is meant for a student.

Does Naishtika Brahmacharya also involve 'Agni-tyaga'?

No, because after all it is a stage of studentship (Brahmacharya). So, he has to continue performing his duties as a student.

living at the teacher’s house, the life of a sage, and that There are
four orders of life:* the householder’s life, of a forest hermit. If
a man remains steadfast in any of these, he attains bliss. common
prerequisite for all is to live at the teacher’s house following one’s
initiation, 4 and all are required not to abandon vedic learning.
After he has learnt the rites, he may undertake the order that he
prefers

Following the rules of a novice student,* a student should serve
his teacher until death, leaving his body in his teacher’s house.

Apasthambha Dharma Sutras 2.22.4

So, he has to behave like a student (Brahmachari) all throughout his life and thus he can not abandon fire.

More references now from Manu Smriti:

2.247. (A perpetual student) must, if his teacher dies, serve his son (provided he be) endowed with good qualities, or his widow, or his
Sapinda, in the same manner as the teacher.

2.248. Should none of these be alive, he must serve the sacred fire, standing (by day) and sitting (during the night), and thus
finish his life.

Daksha Smriti says:

Two classes of Brahmacharin have been mentioned by the wise in the
Smriti. The first is Upakurvanaka (a Brahmana, in a state of pupilage,
who wishes to pass on to the state of a householder); the second is
Naishthika (one who leads a life of perpetual celibacy. (8)

Thus, the similarity that they have with Sannyasis is that they should follow a perpetual celibacy till death.

But, absolute renunciation, like is done in Sannyasa, is not for them as said in the following two Harita Smriti verses:

A Brahmna, whose tongue, generative organ, belly and hands, have all
been controlled, should, resorting to Sannyasa (renunciation), lead a
celibate life near the same preceptor, as long as he lives ; in his
absence near his (preceptor's) son ; [in the latter' s absence,] near
his disciple or in his family. Neither marriage nor [absolute]
renunciation is laid down for the Naishthika* (13,14)

Similarly Usana Smriti says:

If a twice-born person wishes to lead the life of a perpetual
religious student at the house of his preceptor, he should serve him
diligently and assiduously till the destruction of his body. (83)

Or (i.e in the absence of his preceptor), going to a forest he
should duly offer oblations to Fire. And being self-restrained he
should ever and anon study the Spiritual Science. (84)