What happened to the body of elephant whose head was attached to Lord Ganesha and what is the name of the place where the head of Lord Ganesha fall?
Chapter 12 of the Ganapati Khanda of Brahma-Vaivarta Purana describes how Vishnu replaces Ganesha's head with the head of an elephant.
This is a version of the story where Ganesha's head was burnt off by the gaze of Shani. (As for the original head of Ganesha this story says the head was absorbed into the body of Krishna in Goloka)
शनेश्व दृष्टिमात्रेण चिच्छिदे मस्तक मुने। चक्षुर्निमीलयामास तस्थौ नम्राननः शनिः॥७॥
O sage, at the gaze of Saturn, the head of the child was cut-off and Saturn closed his eyes at once, looking downwards and stood there.
तस्थौ च पार्वतीक्रोडे तत्सर्वाङ्गं सलोहितम्। विवेश मस्तकं कृष्णे गत्वा गोलोकमीप्सितम्॥८॥
मूच्छा संप्राप सा देवी विलष्य च भृशं मुहुः। मृतेव च पृथिव्यां तु कृत्वा वक्षसि बालकम्॥९॥
विस्मितास्ते सुराः सर्वे चित्रपुक्तलिका यथा। देव्यश्च शैला गन्धर्वाः सर्वे कैलासवासिनः॥ १०॥
तान्सर्वान्मूर्च्छितान्दृष्ट्वैवारुह्य गरुडं हरिः। जगाम पुष्पभद्रां स चोत्तरस्यां दिशि स्थिताम्॥ ११॥
The head of the child who was lying in the lap of Parvati was severed and his body was soaked in blood. The severed head of the child went to Goloka and entered into the body of lord Krsna.
Parvati started lamenting and carrying the child in her lap she fainted and fell down on earth like dead. All the gods were surprised and were flabbergasted. They stood there like puppets. All the gods, the goddesses, the mountains, Gandharvas and all the dwellers of Kailasa behaved similarly.
Thereafter, finding all the people fainted Vişņu mounted on Garuda, went to the northern direction and reached the bank of Puspabhadra river.
Vishnu severs the head of a male elephant that is resting surrounded by its wives and children.
पुष्यभद्रानदीतीरे ह्यपश्यत्कानने स्थितम्। गजेन्द्रं निद्रितं तत्र शयानं हस्तिनीयुतम्॥ १२॥
तथोदक्छिरसं रम्यं मूर्च्छितं सुरतश्रमात्। परितः शावकान्कृत्वा परमानन्दमानसम्॥ १३॥
शीघ्र सुदर्शनेनैव चिच्छिदे तच्छिरो मुदा। स्थापयामास गरुडे रुधिराप्तं मनोहरम्॥१४॥
Reaching the bank of Puspabhadra river, he found an elephant enjoying the company of cow elephants.
He was resting after enjoying the company of the cow elephants, placing his head toward the north. The elephant-calves surrounded him.
Lord Visnu delightfully cut-off his head by using Sudarsana-cakra which was soaked in blood and kept it over Garuda and was quite pleasant to look at.
The wives and children awaken and begin praying to Lord Vishnu. Becoming pleased with them Vishnu restores the male elephant back to life and bestows a boon of long life upon the elephant family.
गजच्छिन्नाङ्गविक्षेपातप्रबोर्ध प्राप्य हस्तिनी। शावकान्बोधयामास चाशुभं वदती तदाः।।१५।।
रुरोद शावकैः साद्ध सा विलप्य शुचातुरा। तुष्टाव कमलाकान्तं शान्तं सस्मितमीश्वरम्।।१६।।
शंखचक्रगदापद्मधरं पीताम्बरं परम्। गुरुडस्थं जगत्कान्तं भ्रामयन्तं सुदर्शनम्॥ १७॥
With the severing of the head of the elephant, the cow elephants were awakened and getting panicky made the young elephants to wake up.
Thereafter the cow elephants with their progeny offered prayer to lord Visnu, the husband of Kamala, who was peaceful, wearing a smile on his face holding sankha, cakra, gadā and padma in his four hands, clad in yellow garments, mounted on Garuda, who was the lord of the entire universe and was wielding the Sudarsana cakra.
निषेकं खण्डितुं शक्तं निषेकजनकं विभुम्। निषेकभोगदातारं भोगनिस्तारकारणम्॥ १८॥
She said, "You are competent to shatter the birth, the creator of the birth, glorious, bestower of pleasure, emanating from the birth and the one who relieves the people from the worldly pleasures, being the soul cause of the same."
प्रभुस्तत्स्तवनातुष्टस्तस्यै विप्र वरं ददौ। मुण्डातुण्डं पृथक् त्य युयुजेऽन्यगजस्य च॥ १९॥
जीवयामास तं तत्र ब्रह्मज्ञानेन सर्ववित्। सर्वागे योजयामास गजस्य चरणाम्बुजम्।।२०।
त्वं जीवाकल्पपर्यन्तं परिवारै: समं गज। इत्युक्त्वा च मनोयायी कैलासं ह्याजगाम सः॥ २१॥
O Brahmana, lord Visnu got pleased with the prayer and getting satisfied granted a boon. He lifted the head of the elephant from the trunk. With the application of the divine knowledge, he brought back to the dead elephant life.
The lord who is all knowledgeable, touched the body of the elephant with his lotus-like feet and said, "O elephant, you will remain alive with your family for a kalpa."
Thus speaking lord Visnu reached the peak of the mount Kailasa.
आहृत्य पार्वतीहस्ताद्बालं कृत्वा स्ववक्षसि। रुचिरं तच्छिरस्सम्यग्योजयामास बालके॥२२॥
He took out the trunk of the child from the hand of Parvati, kept him in his lap and joined the elephant head with the trunk of the child.
There are various stories about how Ganesha got the elephant head but the most comprehensive one I have encountered states that it originated from Gajasura.
To put it briefly, Gajasura asked Shiva for everyone to remember his head after death, to which Shivji responded affirmatively by releasing him from the samsaric wheel, taking his head to be later mounted on Ganesh.
The whole episode was taking place at Mount Kailash, but I don't know of any specific information regarding elephant's body after liberation.
For the sake of preservation (in case linked pdf becomes unavailable), I will quote the whole story about Shiva and Gajasura that resulted in Ganesha obtaining the latter's head.
Once, there lived an Asura (demon) with all the characteristics of an elephant, called Gajasura. He was an ardent devotee of Lord Shiva. Gajasura has undergone a severe penance (or tapas) for many years to receive special boons. Lord Shiva was pleased by this austerity, and decided to grant him a boon. The demon wished that he could emanate fire continually from his own body so that no one could ever dare to approach him. The Lord granted him his request. Gajasura was not satisfied with that, and continued his penance. Lord Shiva appeared again in front of him. This time the demon asked Shiva to reside in his stomach. Shiva granted his request, and he took up residence in Gajasura’s belly immediately.
Lord Shiva is also known as Bhola Shankar because He is the God, who satisfies easily, and grants boon to His devotees whatever they desire without regard for the consequences. From time to time, this created complicated situations, and put Himself into some trouble. Meanwhile on Mount Kailash, Goddess Parvati, Lord Shiva’s wife, was worried about Lord Shiva’s whereabouts. She searched for Him everywhere without any luck. As a last resource, she went to Lord Vishnu, requesting him to find her husband. He, who knows everything, assured her that Shiva was fine, and asked her to go back to Kailash.
Then Vishnu, the wise director of the cosmic game, staged a small play. He changed his appearance into a flutist, transformed Shiva’s vehicle Nandi (the bull) into a dancing bull, and performed a wonderful dance in front of Gajasura. The enchanting performance of the bull sent the demon into intense joy, and he asked the flutist to tell him what he desired. The musical Vishnu responded: "Can you give me that which I ask?" Gajasura replied: "Who do you take me for? I can immediately give you whatever you ask." The flutist then said: "If that's so, liberate Shiva from your stomach." Gajasura understood then that this must have been no other than Vishnu himself, the only one who could have known that secret and he threw himself at his feet. Having liberated Shiva, he asked him for one last gift: "I have been blessed by you with many gifts; my last request is that everyone remembers me adoring my head when I am dead." To fulfill this wish, Lord Shiva severed Gajasura’s head and freed him from the cycle of birth and death. He carries the head along with him.
Parvati at Kailash came to know of Lord Vishnu’s victory and her husband’s return. She was very happy, and wished to take a bath. She created a boy out of the dough that she uses in her bath, and gave a life to guard the entrance of her house. She asked him not to allow anyone inside. After saying this, she went inside to take the bath. Meanwhile, Lord Shiva returned home. When he tried to enter the house, the boy stopped him at the door as he was unable to recognize his father, Lord Shiva. In anger, Shiva cut off the boy’s head with his trident (Trishul). Parvati was unaware of what had happened and received Lord Shiva with warmth, for he had returned after a long period. During the course of their conversation, Shiva mentions the incident at the palace gates and tells her about severing the child’s head. When Parvati heard about that, she burst into tears. Shiva also felt sorry for his anger. In order to console Parvati, Shiva sent out his troops (Ganas) to search for the boy’s head. When they couldn’t find the head, Shiva attached the boy’s head with that of Gajasura, and revived him. After looking at the child (Ganesha), Parvati asked Shiva that who will consider this god with the face of an elephant. Shiva blessed the child, Ganesha that everyone would worship Ganesha before beginning any work, and people will worship him first before any other God in any kind of undertaking.