I had been planning to sit back and see what ideas came, but since the question may be under threat I might as well post my thoughts now. Eat, drink and be merry. Tomorrow it'll be a ruin...
Celebration of the Eldest. Each of the Qim (slang for the believers) know that he will outlive the others. But in the Church he sees today, someone else is the Eldest. If each of them is prepared, in the first few decades or centuries of his life, to defer to the Eldest, then when he is Eldest, all in the Church will defer to him. Note that this deference only applies to the very oldest person in the entire Church, because from his own perspective the second-eldest will see that person die very quickly. This certainly gives the Church the appearance of a cult of the aged to the casual observer.
Quest for Disproof. The Church of Quantum Immortality is not a normal church. The Qim don't like their philosophy - they simply believe in it. Many of the followers have escaped freak accidents or lived a long time. They have started to become convinced that they won't die. But they still want to believe that they will have a normal death, and not suffer endlessly. So they back research meant to discredit the ideas of their faith, and periodically reject the conclusions as uncompelling.
Quest for Life Extension. The Qim believe they will live forever, so they want to do so more comfortably. Therefore, they generously support every David Sinclair type working on research that is proposed to rejuvenate cells and give people the benefits of youth again. They canvas the neighborhood looking for donations with unusual fervor - again confirming their cult-like status to the outside observer.
Quest for Endurance. As the Qim believe that great suffering awaits them, they travel the world, sharing the stories of fakirs and shamans and metastatic cancer patients, studying the doctrines of other religions such as Buddhism that address the issue of suffering, attempting to gain knowledge that will help them while they are yet able. The gruesome festivals where some of these members share their insights have confirmed the group as a cult beyond all reasonable doubt.
Heresy of Division. Some of the Qim believe that a person, though immortal, can be divided by severing the corpus callosum and other nerve bundles. They assert that such divisions will happen spontaneously until each neuron is a separate organism, existing at too low a level of consciousness to experience pain. This has led to some of the more widely publicized horrors one reads in the news.
Heresy of Union. Some of the Qim have split off to a rival sect with minimalist Hindu trappings, declaring that the Universal Atman is immortal but that the individual consciousness is not - in other words, that a person does not die so long as some other person exists instead; that this is merely a loss of some memories. The Church generally opposes and derides them with Pascalian wager arguments, and sometimes more disturbing methods.