Does the relinquishing of views prohibit belief in the permanence of buddha nature?

The idea of 'permanence' is future-bound. Permanence carries the thought: "That which now is will be so at every point in the future, without exception." That is an illusion: an act of imagining the world as we would like it to be, ignoring that the world will inevitably disappoint us. It generates dukkha.

Buddha nature is not time-bound. It only exists in the present, can only be experienced in the present, and only understands past and future as aspects of the present. People often mistakenly call it 'eternal' or 'permanent', but what they mean to imply is that it is timeless.

When we relinquish all views we give up the worry that it might be better to be somewhere else or somewhen else. It might be, but we are here and now (in Buddha nature) and all movement starts from there.

In the famous parable of the raft, the Buddha says that his Dharma is, ultimately, to be abandoned.

"You, myriad monks, should know that my expounding of the Dharma is like in the parable of the raft. Even the Dharma will be abandoned. How much more so that which is not the Dharma?"

(Diamond Sūtra T235.749b7)

There is another time when the Buddha describes a teaching that is to be abandoned, namely when he outlines "right view with effluents" as opposed to "right view that is noble" and contrasts these both with "wrong view." From the Pāli suttas:

"And what is right view? Right view, I tell you, is of two sorts: There is right view with effluents, siding with merit, resulting in acquisitions [of becoming]; there is right view that is noble, without effluents, transcendent, a factor of the path.

"And what is the right view with effluents, siding with merit, resulting in acquisitions? 'There is what is given, what is offered, what is sacrificed. There are fruits & results of good & bad actions. There is this world & the next world. There is mother & father. There are spontaneously reborn beings; there are contemplatives & brahmans who, faring rightly & practicing rightly, proclaim this world & the next after having directly known & realized it for themselves.' This is the right view with effluents, siding with merit, resulting in acquisitions.

"And what is the right view that is noble, without effluents, transcendent, a factor of the path? The discernment, the faculty of discernment, the strength of discernment, analysis of qualities as a factor for awakening, the path factor of right view in one developing the noble path whose mind is noble, whose mind is without effluents, who is fully possessed of the noble path. This is the right view that is noble, without effluents, transcendent, a factor of the path.

"One makes an effort for the abandoning of wrong view & for entering into right view: This is one's right effort. One is mindful to abandon wrong view & to enter & remain in right view: This is one's right mindfulness. Thus these three qualities — right view, right effort, & right mindfulness — run & circle around right view.

(Mahācattārīsakasutta MN 117 translated by Venerable Ṭhānissaro)

So we have established that what is not the Dharma is to be abandoned, just as wrong views are abandoned for right views in the Mahācattārīsakasutta. But what does it mean now for "the Dharma" to be abandoned (including, presumably, the teachings on "buddha-nature" alluded to by the OP)?

Does this mean we have to abandon all right views?

My answer to this is a cautious "No" that I hope I can satisfactorily contextualize.

In the parable of the raft, the raft of the Dharma is abandoned at the far shore. It is not abandoned amidst the sea, leaving the seafarer to die swimming in the treacherous waters of saṃsāra. If we are to abandon "right view" at all, be that right view toward the aggregates, toward the dharmas, or toward the Buddha's nature, it is not "amidst the sea," so to speak. By the time that it comes for "us" to abandon even the Dharma, there is no more "us" to abandon it. "We're" at that far shore, with no "we" present.

“Then, when he had got across and had arrived at the far shore, he might think thus: ‘This raft has been very helpful to me, since supported by it and making an effort with my hands and feet, I got safely across to the far shore. Suppose I were to hoist it on my head or load it on my shoulder, and then go wherever I want.’ Now, bhikkhus, what do you think? By doing so, would that man be doing what should be done with that raft?”

“No, venerable sir.”

(Alagaddūpamasutta MN 22 translated by Venerable Bodhi)

So why, when we know that the Dharma is for crossing and not for grasping, might some people be moved to say things to effect of "The sea cannot be crossed while grasping a raft?" Why do some reduce the Dharma to "a collection of views?"

IMO, one of the sources of this misunderstanding is a misreading of the Abyākatasutta at AN 7.54. There, we read:

“Because of the cessation of views, monk, uncertainty doesn’t arise in an instructed disciple of the noble ones over the undeclared issues. The view-standpoint, ‘The Tathagata exists after death,’ the view-standpoint, ‘The Tathagata doesn’t exist after death,’ the view-standpoint, ‘The Tathagata both does and doesn’t exist after death,’ the view-standpoint, ‘The Tathagata neither does nor doesn’t exist after death’: The uninstructed run-of-the-mill person doesn’t discern view, doesn’t discern the origination of view, doesn’t discern the cessation of view, doesn’t discern the path of practice leading to the cessation of view, and so for him that view grows. He is not freed from birth, aging, & death; from sorrows, lamentations, pains, distresses, and despairs. He is not freed, I tell you, from suffering & stress. But the instructed disciple of the noble ones discerns view, discerns the origination of view, discerns the cessation of view, discerns the path of practice leading to the cessation of view, and so for him that view ceases. He is freed from birth, aging, & death; from sorrows, lamentations, pains, distresses, and despairs. He is freed, I tell you, from suffering & stress.

(Abyākatasutta AN 7.54 translated by Venerable Ṭhānissaro)

I think that among some people who are more familiar with the suttas than the Dhamma, there is a tendency to just read the first sentence of the Buddha's response and ignore the vital context of the sutta.

For instance, when we read a Pāli sutta, we have the opening question of the sutta (generally posed by someone other than the Buddha) and the opening elaboration by the Buddha. After these two, the rest of the sutta will necessarily follow this exposition. Not only suttas, but sūtras also do this. Sūtras like the Flower Garland, the Lotus, and Prajñāpāramitā open with a question that the entire sūtra wishes to answer -- and these are compendious Mahayana vaipulyas, not short EBT suttas. If the vaipulya in question does not begin with such a question, the question that it does begin with will be treated as if it contains within it the seed for all of the material presented in the sūtra.

What is our "central question" that informs the exegesis of AN 7.54? It is

“Lord, what is the cause, what is the reason, why uncertainty doesn’t arise in an instructed disciple of the noble ones over the undeclared issues?”

This sutta, complete with its section on the cessation of view, etc., is dealing with particular views. In particular, it deals with four views that it outlines in the body of the sutta. 1) That ‘The Tathagata exists after death,’ 2) that ‘The Tathagata doesn’t exist after death,’ 3) ‘The Tathagata both does and doesn’t exist after death,’ and 4) ‘The Tathagata neither does nor doesn’t exist after death.’ It is due to the cessation of these views that "uncertainty doesn't arise," to use the language from the sutta directly. So we can see that the Abyākatasutta does not actually instruct the followers of the Buddha to abandon all views in toto, including the right ones. This is, however, a common way to read the sutta. It is, IMO, misconceived in how it is presented when it is presented like that.

This still doesn't address the issue of the eventual abandonment of right view, "buddha-nature" potentially along with it, once we are at the far shore. To address this, there is a quotation from Grandmaster Zhìzhě that has always been close to my heart. This is from his magnum opus, Mahāśamathavipaśyanā, and to this day it greatly informs the meditative curriculum of seminarians of the Japanese Tendai priesthood (as well as being a greatly influential meditation treatise in China, Korea, Vietnam, etc., among monastics and laity alike):

In order to enter into emptiness, you must contemplate conventional existence; emptiness is realized through this encounter. This realization is as when the clouds are scattered and vanquished, and above is made manifest and below is clear [...] The conventional is that which is to be destroyed; the real is that which is used to destroy [...] If you enter emptiness, you realize that emptiness itself has no being, and re-enter the conventional with that insight, knowing that this contemplation is done for the sake of saving sentient beings, and knowing that the real is not reality but a utility that appears conventionally. Therefore we say "entering from emptiness" [從空入], and one who attains this contemplation differentiates the proper medicine according to the disease without mistaken discriminations.

(Mahāśamathavipaśyanā T1911.23c12, translated by Paul Swanson and published as "Clear Serenity, Quiet Insight," p. 452)

What the worldling knows as "reality" is in fact delusion. "Reality," to the Buddhas, is a utility, a utility used for the saving of sentient beings. To save sentient beings, sometimes they say "the Buddha's nature" or sometimes they say "no nature." Sometimes they say, "All beings have the buddhadhātu," and sometimes they say, "The buddhadhātu is limited to a gotra of certain persons." In the Mahāyāna, the ekayāna teachings are like the Buddhas who teach of the universal buddhadhātu. The pañcagotra teachings are like the Buddhas who say that "Buddhahood" is a specific gotra.

In crossing over to the far shore, in becoming a Buddha, in order to say "no nature" when the utility is required, you must be able to abandon "the Buddha's nature." This is easy because, after all, the Buddha's nature is "no nature."

The many Buddhas have immeasurable implementations of appropriate methodologies, and the many phenomena have no fixed natures. To liberate beings, they say "All is real," or they say "All is unreal," or they say "All is real and unreal," or they say "All is neither real nor unreal." "All is real," because, inquiring into the true aspect of all phenomena, each and every enters into the highest truth of equality and oneness of characteristic, which is to say with no characteristic. It is like the many rivers of different colours and different tastes entering into the great ocean of the same colour and the same taste.

(Madhyamakaśāstra T1654)

At the end of the day, "the Buddha's nature," just like the notion of and appearance of "the Buddha," is a pointer signifying the Dharma and is not the Dharma itself. Were it the Dharma itself, you could enlighten someone simply by saying "You have the Buddha's nature." Certainly, for some people, they might have such an extraordinary experience. The pointer was good for them. But most people will just say, "That's awfully nice, but what does it mean for me to have the Buddha's nature?" "Buddha-nature" signifies something external to what it is as "a word" or "a name," and that signified thing is arrived at outside of the paths of body, speech, and mind. As such, it is not discarded like a view, nor is it adopted like a view. IMO.