Getting the card dropped my credit score within the categories of 'Age of Credit' and 'Credit Inquiries'. I couldn't imagine constantly getting and dropping cards.
I've applied and been approved for 16 cards in the past 18 months. My credit score is somewhere between 770-810, depending on which bureau is reporting. The number of credit inquiries obviously lowers my score slightly, but it has a low impact and will gradually drop off (~2 years for each card). Similarly my "Age of Credit" is lowered by applying for more cards, as the newer cards reduce the average. I already had open accounts with 10-18 years of history, so its effect wasn't as pronounced as someone who might be recently establishing a credit history.
When I stop churning, my scores will gradually increase (probably ending
higher than when I started churning), but my scores are already high enough that I don't have any issues, i.e. I've never had a credit application declined.