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.