Halt! Show Me Your Virus Papers
These documents would certify that the passport-holder had mounted a positive immune response to the virus and was presumed to be resistant to further infection. Passport-holders might safely be released from public health restrictions and allowed to return to work or study, and to live a more normal everyday existence.
For most of the nineteenth century, immunity to yellow fever divided people in New Orleans, Louisiana, between the ‘acclimated’ who had survived yellow fever and the ‘unacclimated’, who had not had the disease. Lack of immunity dictated whom people could marry, where they could work, and, for those forced into slavery, how much they were worth. Presumed immunity concentrated political and economic power in the hands of the wealthy elite, and was weaponized to justify white supremacy.
Immunity passports might also create what economists call perverse incentives. If people can't or won't get vaccinated, they may have an incentive to become infected as a means of getting their passport. A market for fake passports will inevitably spring up, just as it has for so-called mask exemption cards. You can buy those on Amazon.
And of course, these days no controversy is complete without accompanying conspiracy theories. Anti-vaxxers will view this as another attempt to coerce or punish members of their movement. Some Christian groups see immunity passports a "mark of the beast" and a sign of the end times. Libertarians like Ron Paul say it's just another way for government to achieve greater power over free individuals, and fret about what other personal information might be stored on this "passport."
Some people argue that the idea is worth considering, and that there are ways to mitigate the negative effects. But this is a minority opinion. The public health community seems to be in agreement that immunity passports are only likely to further divide us and risk amplifying problems in society.
And we have enough problems as it is.
Update 1/24
Today's New York Times has a story about how the kinds of divisions mentioned above may be occurring even without passports. For example a luxury travel agency is offering travel to the United Arab Emirates to be inoculated with "privately obtained" vaccine, with travel adventures for the newly immune after that.