What is to be done when worshippers push someone to lead the prayer or the call to prayer, while he feels fear of showing off, though he sometimes feels happy about it?

Fear of showing off (riya') is commendable and should not lead one to abandon good deeds. It is also recommended to step forward to lead people in prayer due to the great reward it entails. The joy a Muslim finds when reciting the Quran or the Adhan is praiseworthy and indicates complete faith, so long as it does not reach the level of blameworthy self-admiration.