The older religion gave way across Polynesia within a few generations of sustained European contact, and the process is often told too simply.
In Hawaii the kapu system was overthrown from inside. In November 1819, months before the first American missionaries arrived in April 1820, Kamehameha II ate publicly with his mother Keopuolani and with Kaahumanu in deliberate breach of the eating kapu, and ordered the heiau dismantled and images burned. Resistance was defeated in battle at Kuamoo. The reasons argued by historians include the erosion of the system by years of contact with foreigners who broke it without visible consequence, the political interest of chiefly women excluded by it, and disruption from disease and trade.
In Tahiti the London Missionary Society landed in 1797 and made almost no progress for fifteen years. The turn came through Pomare II, who allied with the missionaries for political and material reasons, won a decisive battle in 1815, and was baptised in 1819; conversion of the population followed his. Elsewhere (Tonga, Samoa, the Cook Islands, Aotearoa, and later), Christianity spread through similar chiefly conversions and, very substantially, through Polynesian missionaries sent from island to island.
Introduced disease shaped all of this. Populations fell catastrophically in many islands, and priesthoods whose authority rested on securing wellbeing were discredited by their inability to stop it.
Polynesia is now heavily Christian, with high church attendance and churches central to community and diaspora life. But the older material was not erased. Genealogies, chants, place names, carving and protocol persisted, sometimes inside the churches. From the 1970s the Hawaiian cultural renaissance and parallel movements elsewhere revived language, hula in its ritual forms, and long-distance navigation, marked by the voyage of the Hokulea from Hawaii to Tahiti in 1976 without instruments. Some heiau and marae have been restored and rededicated, and traditional protocol is used at public occasions. Practitioners today range from Christians who regard mana and tapu as cultural inheritance to a smaller number reviving the older worship directly.