One of the first quirks of Classical Arabic orthography one learns is the alif maqṣūrah or 'Broken Alif', this is, in modern orthography a word-final yāʔ without two dots below it, which you are to pronounce as an ā, e.g. ḥattā 'until, even' حتى <ḥtỳ>, kubrā 'bigger/biggest (fem.) كبرى <kbrỳ> and ramā 'he threw' رمى <rmỳ>. While this sign is graphically distinct from the y now, it originally was not, and was not in the Qurʔān. As you might expect, this spelling is a historical spelling. These words originally contained a y, which was lost at some point in the history of Arabic.
At some point in the history the word-final sequence -ay (or -aya, depending on when you think the word-final a was lost) shifted to -ā, but the spelling was never adjusted for this. A similar development happened with word-final -aw(a), as in *ʔatawa 'he came' > ʔatā, but for the w, this development before the establishment of the orthography, and therefore the verb is simply written with a final alif and not with a final waw: اتا <ʔtʔ>. The form with final w is attested in Pre-Islamic Arabic.
In Classical Arabic orthography, whenever the ā written with alif maqṣūrah is followed by a clitic, as for example in ramā-hu 'he threw it', the orthography stops writing the ā with an yāʔ and instead writes it as normally as an alif: رماه <rmʔh>. This is different in the QCT, where the yā remains written رميه <rmyh>. This has lead people to suggest that in the original language of the Qurʔān's composition, the sound that was marked by the alif maqṣūrah in this context was still a diphthong or perhaps a ē (e.g. Thackston 1994: 275).
While it is tempting to allow for this archaic pronunciation to be part of the language of the QCT, I don't think the QCT conclusively points to this if we remain conservative in our approach. It is certainly true that the QCT orthography has a different norm than the latter Classical orthography, but if we accept that word-final <y> can be read as ā in certain archaic spellings, I don't think it's a stretch that this archaic spelling would also be employed word-internally.
There are, however, some reading traditions of the Qurʔān that actually pronounce all cases of ā < *ay(a) as ē, so certainly this pronunciation was of someone's language, but there's no reason to assume that this more archaic reading tradition is the more correct one for the language. Moreover, it should be noted that the readition tradition that Lameen Souag talks about even pronounces some instances as ē where the QCT actually has a final ʔ, namely in šāʔa شا <šʔ>, which originally came from *šayaʔa, so certainly this archaic pronunciation is not fully equal to the QCT orthography. This makes assuming that final y-verbs where pronounced with ē or ay(a) even more problematic.
There is yet another reading tradition, namely the Warsh reading tradition, that has ē in environments that is slightly more akin to the QCT orthography. Verbs that end in final <y> have an ē as well as the original sequence aya in nouns like Tawrāh (in documents that mark the Warsh pronunciation, a dot below the consonant preceding the vowel, marks the pronunciation ē). Interestingly, the prepositions that are written with a word-final <y> like ʔilā الى <ʔly>, ʕalā على <ʕly> and ḥattā حتى <ḥty> are pronounced with an ā. The difference between these prepositions and the verb can be understood from the shape in Proto-Arabic. The verbs of this type originally had a final triphothong /aya/, e.g. *ramaya > ramē whereas the prepositions of this type ended in final /ay/, e.g. *ʕalay > ʕalā . It seems that the dialect of the reading tradition underwent a shift *aya > ē but *ay > ā. 
Finally, if we look at early Qurʔānic codices, such as the Samarkand Codex (1st word on the 8th line, Q7:38) as well as the Ṣanʕāʔ Qurʔān (Q2:193) we find clear examples of the alif maqṣūrah being mistakenly written with an alif, in ḥattā 'until, even'. This suggests that to the scribes of this manuscript, the alif maqṣūrah was clearly pronounced as an [ā]. One wonders if the fact that this word is pronounced with ā in the Warsh reading tradition is related to this.
But other explanations are also possible. For example, ḥattā never appears in an environment where the etymological y resurfaces, unlike with verbs such as ramā which has it reappear in the paradigm: ramaytu رميت <rmyt> 'I threw'.
In summary, the QCT orthography allows for reading final y verbs as ē more easily than the Classical Orthography. However, just because the orthographic practices differ form the later Classical Standard, does not mean it necessarily reflects the actual spoken form. It is therefore possible that, despite forms like <rmyh>, it was still pronounced ramā-hu, and that the <y> was simply spelled etymologically.
We must conclude that there is no convincing evidence in favour of reading these forms either with ē or with ā. The Samarkand Codex shows one etymological *ay written as ā, but in a position where the Warsh reading tradition, which generally has ē would also read it as ā. If we were to find example of y final verbs spelled with a final alif in old Qurʔān manuscripts, that would be good evidence in favour of it being pronounce ā, but without those examples, we just have an absence of evidence for an ē or ā pronunciation.
The vast majority of the modern dialects follow the developments of the traditional reading of the QCT, with the exception that final w verbs have generally merged completely with final y verbs. However, in Yemen in the Jabal Rāziḥ dialect, all cases where QCT has an alif maqṣūrah, the dialect pronounces it as ē, so: ramē, ḥattē, ʕalē, ʔūlē etc. And it does not shift the vowel as in šāʔa to **šēʔ. The dialect therefore closely mirrors the QCT orthography (but not necessarily the QCT language) (Behnstedt 1987: 133-134).
In the comments Ahmad Al-Jallad pointed out a very good argument that clearly swings the reading of final -ay as ay or ē over final ā, namely: Rhyme in the Qurʔān. There are several Surah's or parts of Surah's that rhyme on -ay, these never interchange with final -ā. Some examples are:
Q79:15-32 ending in (rhyme is broken at Q79:33, but continues again on Q79:34-46) :
mūsā, ṭuwan, ṭaġā, tazakkā, fa-taxšā, al-kubrā, wa-ʕaṣā, yasʕā, fa-nādā, al-ʔaʕlā, wa-(a)lʔūlā, yaxšā, baḏā-hā, fa-sawwā-hā, ḍuḥā-hā, daḥā-hā, wa-marʕā-hā, ʔarsā-hā
Now look at the QCT spelling:
<mwsy, ṭwy, ṭġy tzky, ftxšy, ʔlkbry, wʕsy, ysʕy, fnʔdy, ʔlʔʕly, wʔlʔwly, yxšy, bḍyhʔ, fswyhʔ, ḍḥyhʔ, wmrʕyhʔ, ʔrsyhʔ>
In Arabic rhyming schemes the -hā feminine direct object clitic and -hu the masculine direct object clitic are often 'invisible' to the rhyme. All the words preceding the -hā end in a -ay, (which in Classical orthography would actually be written with <ʔ>, rather than <y>). This is a practice already attested in Pre-Islamic Old Arabic, as found in KRS 2453 as discussed in Al-Jallad 2015.
All examples are: Q20:2-13, 15-24, 36-38, 40-84, Q53:1-56, Q70:15-18, Q75:31-40, Q79:15-46 (without 33), Q80:1-10, Q87, Q92, Q93:1-8 Q96:6-14.
The only consistent deviation I see in the Qurʔān, is femine elatives with y as their third root consonant.
Feminine elative in Arabic have the pattern CuCCā, e.g. kubrā 'biggest'. This final ā is from an original ay sequence, Proto-Arabic *kubrayu, as such, it is transcribed with a <y>: كبري <kbry>. The only exception to this are feminine elatives that have y just before this -ay suffix, in those cases it is written with <ʔ>, e.g. ʕulyā عليا <ʕlyʔ>, dunyā دنيا <dnyʔ> etc.
This is interpreted most easily explained as a dissimilatory rule where the sequence *-yay becomes *-yaw and subsequently -yā. This rule is certainly active in the Quranic Consonantal Text, where dunyā is written as in the regular Classical Arabic orthography. However, it is only these dissimilated -ay sequences that show up in these rhymes. This suggests that the language that these rhyme Surah's were composed in did not undergo the dissimilation that is present in the QCT orthography and that dunyā was simply pronounced dunyē or dunyay.
Examples of these kinds of words "breaking" the rhyme are:
Q53:29 dunyā (also Q20:72, Q79:38, Q87:16)
Note: In the Warsh reading tradition these words are actually dotted to be read as ē, which seems to confirm a form as dunyē.
Note: In the Warsh reading tradition, that verse is Q53:28! In the standard Cairo edition, Q53:28 ends at a word that breaks the rhyme: šayʔan شيا <šyʔ>. This is the only exception to -ay rhyming schemes not accounted for. I'm tempted to take the Warsh unsplit form as the original (this also means that the numbering of verses in the Warsh reading is different!).
Note: Likewise dotted for ē in Warsh.
So my new conclusion is that the QCT absolutely provides evidence that the old diphthong -ay was pronounced not as -ā but as ē or simply -ay. Even nouns that have an -ā in the Qurʔānic orthography but have -ay etymologically, dunyā and ʔaḥyā, were probably pronounced with ē. This agrees with the Warsh reading tradition, and deviates from the QCT orthography.
This close match between the Warsh reading tradition and the clues from the QCT make me lean strongly to accepting the pronunciation of Warsh in this aspect as original, ḥattā and ʕalā were probably pronounced with ā, which also explains the spelling variants mentioned.
It is worth adding that in the Jabal Rāziḥ Yemeni dialect simply has dunyā, and thus also here once again corresponds closely to the dialect of the QCT orthography (but not with that of the Qurʔānic language!).
 It is interesting to note that the word matā متى <mty> 'when?' is pronounce matē in the Warsh reading tradition, which suggests a form *mataya. Outside of Arabic, e.g. in Hebrew this form also yields an unexpected reflex if we assume an original *matay, as it it mātāy in Hebrew. These things may be related in some way. Note that other question words also take a final -a, e.g. ʔayna 'where?' and kayfa 'how'.
Behnstedt, P. 1987. Die Dialekte der Gegend von Ṣaʿdah (Nord-Jemen). Wiesbadne: Harrassowitz.
Thackston, W. M. 1994. An Introduction to Koranic and Classical Arabic. An Elementary Grammar of the Language. Bethesda, MD: IBEX Publishers.