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."
यत्र यत्र रघुनाथ कीर्तनं तत्र तत्र कृतमस्तकाञ्जलिम् । भाष्पवारि परिपूर्ण लोचनं मारुतिं नमत राक्षसान्तकम् ।।
अर्थ : जहां -जहां भगवान श्रीरघुनाथकी संकीर्तन होता है, वहां शरणागत मस्तक, जुडे हुए हस्त कमल और नेत्रोंमें भावपूर्ण आनंद अश्रुके साथ उपस्थित होते हैं , ऐसे राक्षसोंका संहार करनेवाले, श्रीहनुमानको हमारा कोटिश: प्रणाम !
Wherever there is chanting of Lord Shri Raghunath, hanuman ji will be present with head surrendered to his aradhya bhagwan shri Ramchandra, folded hands, lotus and tears of emotional joy in the eyes.
Hanuman ashtakam of goswami Tulsidas ji
ये जपन्ति सदा स्नेहन्नाम मांगल्यकारणम्। श्रीमते रामचन्द्रस्य कृपालोर्मम स्वामिनः।। तेषाम् अर्थे सदा विप्र प्रदाता अहम् प्रयत्नतः। ददामि वांच्छितं नित्यं सर्वदा सौख्यमुतमम्।।
Hanuman ji says to Maharishi Agastya:- O vipra! Those humans whoever chants ever auspicious name of my aradhya bhagwan shri Ramchandra ji with love and devotion! I myself gives them eternal happiness and grants them all the wishes
Hanumanji's main role to protect Rambhakts who regularly chants shri ram and give and enhance rambhakti