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.
Yes, there's a direct reference to Śri Caitanya Mahāprabhu as an avatār of Viṣṇu (specifically Kṛṣṇa's) in the Bhaviṣya Mahā-Purāṇa.
Chapter 10 - Pratisarga Parva (Chaturtha Khaṇḍa), Bhaviṣya Purāṇa.
विसर्जेति नरान्भदान्करुणया प्रपाल्य क्षितौ निदेदायितुमुद्भवः परात्परं स्दलीयं पदम्। कलौ दितिजसम्भवाधिव्यथाब्धिसुरमग्नगान्समुद्धर महाप्रभो कृष्णचैतन्य शचीसुत ॥ ३२ ॥ माधुर्यैर्मधुभिस्सुगन्धवदनः स्वर्णाम्बुजानां वनं कारुण्यामृतनिर्झरैरुपचितः सत्प्रेमहेसाचलः । भक्तारभोधरधारिणी विजयिनी निष्कम्पसप्तावली देवो नः कुलदैवतं विजयते चैतन्यकृष्णो हरिः॥ ३३ ॥ देवारातिजनैरधर्मजनितैस्सम्पीडितेयं मही तङ्कुच्याशु कलौ कलेदरभिदं वीजाय हा वर्तते। चन्नाम्नैव सुरारयो विदलिताः पातालगाः पीडिता म्लेच्छा धर्मपराः सुरेशनमनास्तस्मै नपो व्यापिने ॥ ३४ ॥
महाप्रभो, कृष्णचैतन्य एवं शची सुत ! आप ही इस भूमण्डल में मनुष्यों को उत्पन्नकर करुणावश उनका पालन-पोषण करते हैं, तथा सर्वोत्तम अपने उस परमपद को उन्हें प्रदान करने के लिए अवतरित भी हुए हैं। इसलिए इस घोर कलि के समय दिति-पुत्र (दैत्यों) से पीड़ित हम देवों की रक्षा कीजिये । हमारे कलदेव भगवान चैतन्य कृष्ण की सदैव विजय होती रहे, जो सुगंधित सुन्दर पदार्थों की सुरभि से सुरभित अंग, सुवर्ण कमल के वन, तथा प्रेम के इस प्रकार के हिमालय हैं, जो करुणामृत के झरनों से विभूषित और भक्त रूपी मेघों को धारण करने वाली विजयिनी एवं निष्कम्प सप्तावली देवी से युक्त है। हम लोग सुरेश वन्दित एवं (समस्त) व्यापक (उस देव) को नमस्कार कर रहे हैं, जो इस घोर कलि के समय अधार्मिक देवशत्रुओं से पीड़ित पृथिवी को पल्लवित करने के लिए अपने शरीर को संकुचित कर बीज रूप में पुनः अवतरित हैं, और जिसके नाम ही सुनकर सुरारिगण पददलित होकर पाताल पहुँच जाते हैं एवं अधर्मपरायण म्लेच्छगण सदैव पीड़ित होते हैं ।
English Translation of the above Hindi Translation by Self
O Lord! Śrī Kṛṣṇa-Caitanya Mahāprabhu, the son of Devī Śacī. It is you only, who creates the Humans on this Earth, and out of Empathy, protects and sustains them. You have descended here to bestow on those worthy humans your Supreme Abode (Param-Padam). O Lord! Therefore in these times of Kaliyuga, do protect us from the torments of those sons of Diti, the Daityas. May our clan's deity - Śrī Kṛṣṇa-Caitanya be ever-victorious. He who is decorated with the aroma of myriad beautiful items, with the complexion of a Golden-lotus, whose mercy is greater than the great Himalayas, he who is adorned with the everflowing ambrosia of mercy and empathy towards his devotees, and who is one with the Devī Saptāvalī, the ever-victorious and disciplined. We devas, offer our obeisances to that all-pervading - God, who's worshipped by all deities, who have descended on this Kaliyuga stricken Earth to relieve its burden, and by the mere chant of whose name, the Daityas flee to the netherworlds, and the barbarians get petrified.
The same version is used by the Gita Press Gorakhpur in their Abridged Hindi Translation of the Bhavishya Purana.
उस समय रुद्र, वसु, विश्वेदेव, मरुद्रण, साध्य, सिद्ध तथा भास्कर आदि देवों ने उस सनातन हरिरूप बालककी दिव्य स्तुति को ओर कलियुगमें दितिपुत्रोद्रारा पीडित देवताओं तथा अधर्मसे दुःखी पृथ्वीका उद्धार करनेके लिये प्रार्थना की । यही आगे चलकर श्रीकृष्णचैतन्यके नामसे विख्यात हुए।
English Translation of the above Gita Press's Hindi Translation by Self
At that time, to ameliorate the grief of Celestials due to the Daityas, and to relieve the Earth from the burden of sins, all the deities including Rudras, Vasus, Visvedevas, Sadhyas, Siddhas, Adityas, etc., prayed to that infant of the form of Hari, and in the Kaliyuga, that child popularly came to known as Śrī Kṛṣṇa Chaitanya.