I heard that who are living in kashi, they never born after death in kashi, why they never born, is this speciality only for kashi (or) any other city it has, and why their life span of time move fast.
Yes, that's true as per the Puranas and that's what the Hindu belief is. That's why many Bengalis used to build additional homes in Kashi so that they can pass their last days in Kashi. And, that's how Kashi has become the second home of the Bengali people.
Kashi is considered as the chief among all Mokshapuris or the abode/city of Moksha and is considered as the greatest of all Tirthas.
Quoting from the article "Mokshapuri Kashi" hosted at Kamakotimandali:
grhamedhyatra vishwesho bhavAnI tatkuTumbinI |
sarvebhyaH kAshisaMsthebhyo mokShabhikShAM prayacChati ||
`The divine couple residing in Kashi - Sri Vishwanatha and Sri Annapurna, grant the Bhiksha of Moksha to all residents of Kashi' - so say the scriptures. Thus, Kashi has been described as the foremost among the Mokshapuris. Kashi Khanda - which forms the fourth chapter of the Skanda Purana, describes the limitless glory of the sacred capital of Sri Vishwanatha. This post is a humble tribute to my Guru whose foremost desire was to reside in Kashi.
From this Hindupedia page we can get the following verse (which is supposedly also from the Skanda Purana) that says if one dies in Kashi he/she gets Moksha.
53.Places of salvation
Daasanath Abra sadasi , jananath Kamalalaye ,
Kasyanthu maranath mukthi , Smaranath arunachale
Seeing Chidambaram, Being born in Thirivarur, Dying in Kashi and just thinking of Thiruvannamalai (Arunachala) would lead to salvation.
There are other references stating the same.
For example, from this answer, we get verses from Shiva Purana. Lord Shiva speaks the Taraka mantra (OM) to liberate beings who die in Kashi.
To understand the meaning of Pranava is to understand me. Pranava is the seed of all the lores.
It shall be understood as very subtle but possessed of great meaning even as the seed of the Banyan tree though very small contain» a huge tree. It is the initial mantra and the essence of the Vedas. Particularly it has me for its form.
I am the the lord beyond attributes, the omniscenet and the omnipotent. I am Siva pervading ail but stationed in the single-syllabled mantra Om.
They say that whatever object there is whether synthesized into one or analysed in pieces is thc meaning of Pranava, thanks to the combination and importance of the attributes.
It is the imperishable Brahman, the means of attaining all objects. Siva creates universe at the outset saying "0m".
- Since there is not much difference bctwecn thc sense and the sound 0m, this is explained thus. Siva is Pranava or Pranava is Siva.
- The brahminical sages, the scholars who realize the identity between the sense and the sound know me as the single-syllabled 0m.
- Hence he who aspires for salvation and is free from aberration shall understand Pranava the cause of all and me as the Nirguna Paramesvara.
- O goddess, I shall give this crest-jewel of mantras at Käsi for the liberation of all Individual souls.
Lord Shiva's words to Goddess Parvati in Shiva Purana's 2nd Chapter of Kailasha Samhita (Translated by J.L.Shastri)
NOTE - I have not found these verses myself. Credit goes to @TheDestroyer for notifying me about them.
Also, Ramakrishna Paramhamsha have actually witnessed Lord Shiva and the Goddess liberating the persons who died in Kashi during his visit to the Manikarnika burning ghat.
The party entered holy Benares by boat along the Ganges. When Sri Ramakrishna's eyes fell on this city of Siva, where had accumulated for ages the devotion and piety of countless worshippers, he saw it to be made of gold, as the scriptures declare. He was visibly moved. During his stay in the city he treated every particle of its earth with utmost respect. At the Manikarnika Ghat, the great cremation ground of the city, he actually saw Siva, with ash covered body, and tawny matted hair, serenely approaching each funeral pyre and breathing into the ears of the corpses the mantra of liberation; and then the Divine Mother removing from the dead their bonds. Then he realized the significance of the scriptural statement that anyone dying in Benares attains salvation through the grace of Siva.
The Gospel of Sri Ramakrishna translated by Swami Nikhilananda, Introduction
Above portions are taken from Pradip Gangopadhaya's answer.
And, I have no idea if there are other cities too having this feature or not. I only know about this belief about Kashi that if one dies there he (or she) gets freedom from the cycle of birth and death (i.e. he gets Moksha).
According to Vedanta schools, upAsanA of the Parabrahman is the only path to mokSha. However several scriptures attribute the fruit of no rebirth to specific actions.
E.g. In Bhagavad Gita chapter 4, the Lord declares that one who understands the secret behind his incarnations sheds his body and is not born again.
Now such points (dying in Kashi, understanding the incarnations, etc.) need to be reconciled with "Parabrahma-upasana" being the only upAya to mokSha.
In the Ramanuja tradition, the purvacharyas have opined that such statements are not to be taken literally. That is, the act specified in such statements would not directly cause the doer to attain mokSha, but would lead the doer towards ultimately pursuing a path that could directly lead to mokSha.
E.g. in the case of dying in Kashi:
One doesn't know when they are going to die. This means they need to go an live in Kashi and wait for their death which may happen in a few days or after a couple of decades.
Living in Kashi means living in the vicinity of temples, vedic scholars, and sattvik people. This would automatically cultivate sattva guNa in the person living in Kashi and impel them towards more spirituality. Ultimately it would lead the person towards Parabrahma-upasana which would ultimately lead towards mokSha.
While the proximal cause behind achieving mokSha is Parabrahma-upasana, one cannot deny that a distant cause for the same is the fact that this person moved to Kashi and living there resulted in his Parabrahma-upasana. This is the meaning behind such statements (in the Ramanuja tradition).
Note: I asked a similar question to my Guru in the year 1998 and I am providing the answer I received. My Guru told me that this topic is dealt with in detail by Ramanujacharya but I haven't looked around to find the exact source.
Update: One source for the above explanation is from Vedanta Desika's Srimad Rahasyatrayasaram Chapter 6: