The world would end in nuclear holocaust right after the post-cogs emptied our bank accounts
Just a few of the things post-cogs would destroy.
No Password is safe
Seedy post-cogs would use it to learn our usernames and passwords. All the post cog has to do is watch some rich person log in to their bank account and clean them out.
Vegas would close down as divorce rates rocket upwards
Every wife would be able to get a high-light reel of their husband's "work trip" to Vegas. One in four spouses cheat on their partners, so the courts will be backed-up for years as angry lovers use post-cogs as proof in their divorce case.
Mass incarceration because plea-deals are moot, every cold-case is solved, and everyone has broken the law at least once in their life
No need for plea-deals anymore. Call your post-cog and find out exactly what happened. Make sure to record it for the jury to get the maximum possible sentence
Dude acting suspicious? Call a post-cog! Don't like the guy hanging out on the corner? Call a Post-Cog! Want your rival out of the way? Call a Post-Cog!
Chaos in the streets as all agencies get hacked at the same time
Almost all records are only stored digitally, which thanks to the post-cog hack is no longer trusted (see #1).
Millions of prisoners have their sentences change and will be released today. Every person with any savings has been cleaned out by post-cogs, and no one accepts electronic or paper money anymore because of rampant post-cog thievery. We're back to trading in ounces of gold.
Rioting and looting start as people realize they have no money, but need necessities like bread and water.
Finally nuclear holocaust to stop the post-cogs
Finally, people will begin to realize post-cogs must be stopped at all cost. Several countries have stock-piled nuclear weapons, and still use pass-codes (see #1). A few post-cogs find the nuclear launch codes and poof - no more post-cogs, along with the rest of humanity.