Is “do whatever makes you happy” an Islamic idea?

Doing whatever makes you happy is the way of the followers of Nafs e Ammara, and it is not the way of Islam. Nafs e Ammara indulges in the worldly pleasures and takes the man to the valley of desires, which is tempting yet barren.

Nowadays it is a common practice to say, "Do as you please or do whatever you feel like doing or whatever makes you happy".

Shaykh al-Niffari rahimahullah said, "The Nafs is innately disposed to bad manners, while the servant is commanded to have good manners. Thus, the Nafs by its nature takes along the path of disobedience and the servant turns its back by all his might." [al-Qushayri, al-Risala al-Qushayriyya fi ilm al-Tasawwuf]

For those who put their nafs through rigor in this world, Paradise is promised as a refuge: "But as for he who feared the position of his Lord and prevented the self from inclination, then indeed, paradise will be his refuge" [Qur'an, an-Nazi'at 79:40-41]