I have heard a lot that Lord Hanuman is immortal but I want to know where he lives?
Yes Lord Hanuman is still on this planet. Shrimad Bhagvat Puran Canto 5 Chapter 19 mentions that he lives in the Kimpurush Varsha listening to Ramayan being sung by the denizens of the place and himself joining them in the singing:
किंपुरुषे वर्षे भगवन्तमादिपुरुषं लक्ष्मणाग्रजं सीताभिरामं रामं तच्चरणसंनिकर्षाभिरतः परमभागवतो- हनुमान् सह किंपुरुषैः अविरतभक्तिरुपासते ।
आर्ष्टिषेणेन सह गन्दर्वैरनुगीयमानं परमकल्याणीं भर्तृभगवत्कथां समुपशृणोति स्वयं चेदं गायति ॥
S'rî S'uka said: 'In the land of Kimpurusha the supreme and greatest devotee Hanumân delighting in service at His feet is together with the people always engaged there in worship of the Supreme Lord Râmacandra, the Original Personality who as the elder brother of Lakshmana is so pleasing to Sîtâ.
Together with Ârshthishena [the leader of Kimpurusha] attentively listening to the stories about his most auspicious master and Lordship being chanted by a company of Gandharvas, he [Hanumân] himself prays thus.
Coming to the location of Kimpurush Varsha, it is stated to be situated north of the Himalayas which could refer to the northern range of the mountains since Mansarovar Lake is also associated with it.
Lord Rama told Hanuman to stay on earth on his death time so that he can spread Rama's story and devotion in humans. And Hanuman is following that order of Rama till now. He is spreading Rama Naam in world in many unknown ways and in different forms. As we all know that He can take any form and by this ability of him he participate in devotion process in different form (not in his actual form) so that no one can recognize him. Hanuman's responsibilities on earth are to spread Devotion so that people can be aware of truth about world.
How he is performing his duties assign by Lord Rama to him?
Hanuman is one of the most intelligent Lord (gyaan gun sagar), so he has tons of ways to spread devotion without being noticed by anyone. He does all his work secretly.
Here is an example of his work that he is doing since Rama's death. This video will help you to know that Hanuman is still alive and living in Himalya.
Hanuman is nowhere mentioned in the Vedas. He is not a vedic God. The legend of Hanuman owes its existence due to Valmiki’s Ramayan, where he is introduced in Kishkindha Kand, and then helps Rama. Later day religious treatises, Purana, as well as literature in local languages preached the cult of Hanuman. Tulsidas, gave him much importance, by putting up a story that Hanuman introduced Tulsidas to Rama. The small booklet of forty stanza, the ‘Hanuman Chalisa’ as well as the Sundarkand of this Ramcharitmanas made Hanuman quiet popular.
The sect of Madhwa, a Vaishnava sect, prevalent mostly in coastal Karnataka, in Udipi, and a certain town by the name of Mantralayam, gave much importance to Hanuman. It was because Hanuman was associated with the God of air, the Vayu, as the mythology stated Hanuman to be a son of Vayu. Taking this analogy further, this sect even included the warrior, Bhimsen of Mahabharata, as one of their revered deities, since Bhimsen is also a son of this very deity of Vayu, or God of air, as stated in Mahabharata. Madhwacharya is the third one in this line, as incarnation of the Vayu God.
That is why, it is written, in the monasteries of this sect, that Vishnu is the best in the creation ( he is the creator himself), while Vayu is the best among the living beings हरिः सर्वोत्तमः वायुः जीवोत्तमः.
So, the sect of Madhwa, has to establish Hanuman among the pantheon of vedic Gods, to justify their stand. Vayu is already there, and Hanuman is to be included. For this purpose, this sect has translated a few verses of Vedas, to show that these are dedicated to Hanuman. But since the translation is too perverse, no other sect or scholar accepted this translation.
Then, another scholar, by the name of Neelkanth Chaturdhar, from present day Telangana state of India, happened to be in eighteenth century CE. He wrote a small treatise named as ‘Mantra Ramayanam’ wherein he explained the existence of Rama as the super-most God, as stated in the Vedas. This treatise is based on some one hundred and odd verses from Rigveda, explained by this scholar, in such a perverse manner, to find that these verses are dedicated to Lord Rama. Now, since Rama comes along-with his eternal servant Hanuman, a few verses have been shown to contain references to Hanuman as the servant of Rama. The book has just superimposed Rama on Indra, and Hanuman on the Gods of Marut, to show that the story of Rama is there in the Vedas. Following is the link to this treatise.
This is the only link in Vedas, to this date, that finds Hanuman among vedic deities. But the explanation of the verses is such perverse and misplaced, that this treatise is more considered as laughable, and never taken seriously by any scholar. That is why a copy of the same is not found even with Vaishnava saints of Ramanandi Vairagi tradition, who worship Rama as their deity.
A few other scholars state that the God ‘Vrishakapi’, as stated in Rigveda 10.86, Aitareya Brahmanam 6.29, Gopath Brahmanam 2.6.12, is the same Hanuman. The only argument they forward is that the word ‘Kapi’ used here also means a monkey, hence its link with Hanuman. It is another matter that Vrishakapi means Sun God ( Nirukta 12.17), as well as the soul of a person.