The Gaudiya Vaishnava Sampradaya claim that Chaitanya Mahaprabhu was an incarnation of Lord Krishna. What is the scriptural evidence behind this claim? Is he mentioned in Puranic literature?
First that there is a covered incarnation in Kali-yuga is known from the below verse.
itthaṁ nṛ-tiryag-ṛṣi-deva-jhaṣāvatārair lokān vibhāvayasi haṁsi jagat pratīpān dharmaṁ mahā-puruṣa pāsi yugānuvṛttaṁ channaḥ kalau yad abhavas tri-yugo 'tha sa tvam
In this way, by appearing in various incarnations as a human being, an animal, a great saint, a devatā, a fish or a tortoise, you maintain the worlds, kill those who cause disturbance, and protect dharma in all the yugas. Great Lord! Since you will appear covered in Kali-yuga, you are called Tri-yuga.
That the lord appears in yellow colour can be known from this verse:
āsan varṇās trayo hy asya gṛhṇato ’nuyugaṁ tanūḥ śuklo raktas tathā pīta idānīṁ kṛṣṇatāṁ gataḥ
Your son Kṛṣṇa appeared in previous yugas with three different colors— white, red and yellow—and now he has appeared in a blackish color.
Now, 11th canto describes the yuga-avataras..
It describes the Satya, Treta and Dvapara yuga colors as white, red and black..
And then proceeds to reveal the yuga-avatara of the Kali-yuga.
Since the incarnation is covered or a hidden one as per 7.9.38, the verse reveals indirectly.
kṛṣṇa-varṇaṁ tviṣākṛṣṇaṁ sāṅgopāṅgāstra-pārṣadam yajñaiḥ saṅkīrtana-prāyair yajanti hi su-medhasaḥ
Intelligent persons perform congregational chanting to worship the incarnation of God who constantly sings the name of Krishna. Although his complexion is not blackish, he is Krishna himself who is accompanied by his servants, weapons, and confidential associates.
Here, in the above verses, tviṣākṛṣṇam indicates his complexion is certainly not blackish. His not being blackish indicates a golden complexion based on 10.8.13
The other verses quoted to support avatarhood of Chaitanya Mahaprabhu are:
Vishnu Sahasranama in Mahabharata predicts the golden colored lord:
suvarṇa-varṇo hemāṅgo varāṅgaś candanāṅgadī
In His early pastimes He appears as a householder with a golden complexion. His limbs are beautiful, and His body, smeared with the pulp of sandalwood, seems like molten gold.
sannyāsa-kṛc chamaḥ śānto niṣṭhā-śānti-parāyaṇaḥ
In His later pastimes He accepts the sannyāsa order, and He is equipoised and peaceful. He is the highest abode of peace and devotion.
Mundaka Upanishad 3.1.3:
yadā paśyaḥ paśyate rukma-varṇaṃ kartāram īśaṃ puruṣaṃ brahma-yonim tadā vidyān puṇya-pāpe vidhūya nirañjanaḥ paramaṃ sāmyam upaiti
(When) the seer of brahman sees the golden colored Lord, the controller of all, the creator, the source of unqualified brahman, he attains the highest knowledge and transcends both pious and impious activities, becomes free from worldly bondage, and enters the divine abode of the Lord.
Chaitanya Charitamrita quotes a verse from Upapurana:
aham eva kvacid brahman sannyāsāśramam āśritaḥ hari-bhaktiṃ grāhayāmi kalau pāpa-hatān narān
[The Supreme Lord said to Vyāsadeva] O learned brāhmaṇa, sometimes I accepted the renounced order of life to induce the fallen people of the age of Kalī to accept devotional service to the Lord.
The meaning of each word of the above verses, why the above verses refer to Chaitanya Mahaprabhu is discussed in greater detail in the book Sri Chaitanya Mahaprabhu ki Svayam Bhagavatta by Radha Govinda Nath which is in Hindi
There is no direct reference to Sri Chaitanya Mahaprabhu being an incarnation of Sri Krishna in any Puranas.
However, this verse from the Srimad Bhagavatam is often quoted by Gaudiya Vaishnavas to assert Mahaprabhu's divinity:
yajanti hi su-medhasaḥ
English Translation: In the Age of Kali, intelligent persons perform congregational chanting to worship the incarnation of Godhead who constantly sings the names of Kṛṣṇa. Although His complexion is not blackish, He is Kṛṣṇa Himself. He is accompanied by His associates, servants, weapons and confidential companions. (Srimad Bhagavatam 11.5.32)
This verse has been quoted by Srila Jiva Goswami in his "Krama Sandarbha", and by Krishna Kaviraja Dasa in his "Chaitanya Charitamrita" as well to prove the divinity of Sri Chaitanya Mahaprabhu.