Hanuman had a great role in Ramayana and also took part in Mahabharata of the previous yugas.
Since Hanuman is Chiranjeevi and Brahmachari, he will be there in Kaliyuga and have some role.
So, what is the role of Hanuman in Kaliyuga?
Hanuamn in Kaliyuga is available to us very quick and short span of time. Just chant Ram Nam and he will come to assist you and bless us.
In this age there are plenty of distractions and hence for one to focus on their spiritual growth is rarely possible.
So his role is to bless us and give us victory in whichever good deeds/tasks we do in our life. Hence before starting any tasks we should pray Hanuman.
Jai Shri Ram.
Shri Hanuman ji's role in Kaliyug is:
भूत पिशाच निकट नहिं आवै।
महाबीर जब नाम सुनावै॥ २४ ॥
नासै रोग हरै सब पीरा। जपत निरंतर हनुमत बीरा॥ २५ ॥
संकट तें हनुमान छुड़ावै। मन क्रम बचन ध्यान जो लावै॥ २६ ॥
कौन सो संकट मोर गरीब को,
जो तुमसे नहिं जात है टारो!
बेगि हरो हनुमान महाप्रभु,
जो कछु संकट होए हमारो...
The soul reason Sri Hanuman is alive is for Sri Ram. As described here, from the Uttara Kanda of the Ramayana, After Sri Ram was told by Yama the god of death that his time on earth is over, Sri Ram prepared to go into the Sarayu river along with his brothers Bharat and Shatrughna (as Lakshman was already dead). Before he did so, he gave boons to various people, and in particular this was the boon he gave to Sri Hanuman:
[Sri Ram] said to Hanuman: "It is settled that thou shalt live forever, do thou, now observe thy promise. As long as my history shall run current in this world do thou at my command live happily." Being thus addressed by high-souled Raghav Hanuman attained to great delight and said: "As long as the sacred theme shall pass current in this world I shall live here carrying out thy commands."
Lord Hanuman is still alive nowadays and this is 200% truth!!! Nowadays,he is in cave of Himalayas and his age is 500 or more than that. Actually,this was a boon that Lord Vishnu had granted to Lord Hanuman after Lord Rama's death. I have the picture of the real Lord Hanuman in my house. There is shown the 500 years old Hanuman sitting in the cave of Himalayas,reading a very old book. This is a proof of Hanuman ji. This is my believe that Lord Hanuman is still alive in nowadays!