user5: Question in the title. I'm looking for answers citing either canonical scripture or the position of any particular school of Buddhism.
The inner monologue in the question could also be read as mental chatter; mental noise; superfluous thoughts; extraneous thoughts; or the  unnecessary thoughts that normal people have every moment.
And I mean when the inner monologue has stopped for good; that is, the case when it never comes back.
Another way to read this question might be, Do arahants have an inner monologue?

user295: What stops is the becoming and creation of new fabrications as you are devoid of clinging and craving.

user-1: Yes, householder Anthony, interested,
asavas, effluents or fermentation stop at attaining of Arahat-ship, "the destruction of the asavas". Ones acting, incl. verbalication, is "just" such. Not that there are no more mental verbalisations, but they are no more "schizophrenic", thrown into a "democracy".
Mental verbalications of Noble ones are not personal and have purpose, for example to "hint" toward Devas, to give causes. So 'Do arahants have an inner monologue?', inner is not personal to be understood and there are no "monologues" but shares of compassion.
(Note that this gift of Dhamma is not dedicated for trade, exchange, stacks or entertainment but as a means to make merits toward release from this wheel)

user4773: Anthony's definition of "inner monologue" is clear, but still no quite clear enough.What are these "unnecessary thoughts that normal people have" about?
Assuming we agree that some of these are

thoughts of future
thoughts of past
thoughts arising from conceiving "I am"

The Arahant has none of these:

Atītaṃ nānusocanti,
  nappajappanti nāgataṃ;
  Paccuppannena yāpenti
  They do not sorrow over the past,
  Nor do they hanker for the future.
  They maintain themselves with what is present
http://suttacentral.net/en/sn1.10
Samo visesī uda vā nihīno,
  Yo maññatī so vivadetha tena;
  Tīsu vidhāsu avikampamāno,
  Samo visesīti na tassa hoti;
  One who conceives ‘I am equal, better, or worse,’
  Might on that account engage in disputes.
  But one not shaken in the three discriminations
  Does not think, ‘I am equal or better.‘http://suttacentral.net/en/sn1.20

So the Arahant's mind is considerably quieter than a "normal person".

user18: Thoughts are one of the six sense objects; there is no reason to think that they stop when one becomes an arahant. It is quite clear that both the Buddha and arahants did indeed have thoughts after becoming enlightened. E.g.:

And the knowledge and vision arose in me: ‘Āḷāra Kālāma died seven days ago.’ I thought: ‘Āḷāra Kālāmaʹs loss is a great one. If he had heard this Dhamma, he would have understood it quickly.’
-- MN 26 (Bodhi, trans)

What does stop forever is uddhacca or restlessness.

Restlessness (or agitation) has the characteristic of disquietude, like water whipped up by the wind. Its function is to make the mind unsteady, as wind makes a banner ripple. It is manifested as turmoil. Its proximate cause is unwise attention to mental disquiet.
-- Bhikkhu Bodhi, A Comprehensive Manual of Abhidhamma
In the case of the defilements, [false] view and uncertainty are eliminated by
  the first knowledge. Hate is eliminated by the third knowledge. Greed, delusion,
  conceit (pride), mental stiffness, agitation, consciencelessness, and shamelessness are eliminated by the fourth knowledge.
-- Vism. XXII.65 (Nyanamoli, trans)

The only case in which the inner monologue could possibly stop forever is at the moment of parinibbana, since that is the only event that could be said to be permanent. How else could one understand the word "forever"?
Even if one were to suddenly go without an inner monologue while still alive, there is no reason to infer that the monologue would not start up again; clearly both arahants and Buddhas still have some mental monologue as long as there is still the arising of sankharas.

5: <p>Question in the title. I'm looking for answers citing either canonical scripture or the position of any particular school of Buddhism.</p>

<p>The <em>inner monologue</em> in the question could also be read as mental chatter; mental noise; superfluous thoughts; extraneous thoughts; or the  unnecessary thoughts that normal people have every moment.</p>

<p>And I mean when the inner monologue has stopped for good; that is, the case when it never comes back.</p>

<p>Another way to read this question might be, <em>Do arahants have an inner monologue?</em></p>
 295: <p>What stops is the <code>becoming</code> and creation of new fabrications as you are devoid of clinging and craving.</p>
 -1: <p>Yes, householder Anthony, interested,</p>

<p><a href="http://accesstoinsight.eu/en/dictionary/asava" rel="nofollow noreferrer">asavas</a>, effluents or fermentation stop at attaining of Arahat-ship, "the destruction of the asavas". Ones acting, incl. verbalication, is "just" such. Not that there are no more mental verbalisations, but they are no more "schizophrenic", thrown into a "democracy".</p>

<p>Mental verbalications of Noble ones are not personal and have purpose, for example to "hint" toward Devas, to give causes. So 'Do arahants have an inner monologue?', inner is not personal to be understood and there are no "monologues" but shares of compassion.</p>

<p><em>(Note that this gift of Dhamma is not dedicated for trade, exchange, stacks or entertainment but as a means to make merits toward release from this wheel)</em></p>
 4773: <p>Anthony's definition of "inner monologue" is clear, but still no quite clear enough.What are these "unnecessary thoughts that normal people have" about?</p>

<p>Assuming we agree that some of these are</p>

<ul>
<li>thoughts of future</li>
<li>thoughts of past</li>
<li>thoughts arising from conceiving "I am"</li>
</ul>

<p>The Arahant has none of these:</p>

<blockquote>
  <p><code>Atītaṃ nānusocanti,
  nappajappanti nāgataṃ;
  Paccuppannena yāpenti
  They do not sorrow over the past,
  Nor do they hanker for the future.
  They maintain themselves with what is present</code>
  <a href="http://suttacentral.net/en/sn1.10" rel="nofollow">http://suttacentral.net/en/sn1.10</a></p>
  
  <p><code>Samo visesī uda vā nihīno,
  Yo maññatī so vivadetha tena;
  Tīsu vidhāsu avikampamāno,
  Samo visesīti na tassa hoti;
  One who conceives ‘I am equal, better, or worse,’
  Might on that account engage in disputes.
  But one not shaken in the three discriminations
  Does not think, ‘I am equal or better.‘</code><a href="http://suttacentral.net/en/sn1.20" rel="nofollow">http://suttacentral.net/en/sn1.20</a></p>
</blockquote>

<p>So the Arahant's mind is considerably quieter than a "normal person".</p>
 18: <p>Thoughts are one of the six sense objects; there is no reason to think that they stop when one becomes an arahant. It is quite clear that both the Buddha and arahants did indeed have thoughts after becoming enlightened. E.g.:</p>

<blockquote>
  <p>And the knowledge and vision arose in me: ‘Āḷāra Kālāma died seven days ago.’ I thought: ‘Āḷāra Kālāmaʹs loss is a great one. If he had heard this Dhamma, he would have understood it quickly.’</p>
  
  <p>-- MN 26 (Bodhi, trans)</p>
</blockquote>

<p>What does stop forever is <em>uddhacca</em> or restlessness.</p>

<blockquote>
  <p>Restlessness (or agitation) has the characteristic of disquietude, like water whipped up by the wind. Its function is to make the mind unsteady, as wind makes a banner ripple. It is manifested as turmoil. Its proximate cause is unwise attention to mental disquiet.</p>
  
  <p>-- Bhikkhu Bodhi, A Comprehensive Manual of Abhidhamma</p>
  
  <p>In the case of the defilements, [false] view and uncertainty are eliminated by
  the first knowledge. Hate is eliminated by the third knowledge. Greed, delusion,
  conceit (pride), mental stiffness, agitation, consciencelessness, and shamelessness are eliminated by the fourth knowledge.</p>
  
  <p>-- Vism. XXII.65 (Nyanamoli, trans)</p>
</blockquote>

<p>The only case in which the inner monologue could possibly stop forever is at the moment of parinibbana, since that is the only event that could be said to be permanent. How else could one understand the word "forever"?</p>

<p>Even if one were to suddenly go without an inner monologue while still alive, there is no reason to infer that the monologue would not start up again; clearly both arahants and Buddhas still have some mental monologue as long as there is still the arising of sankharas.</p>
