If ISP A assigned one of their users an IP address in ISP B's network not much would happen aside from that user's internet not working. Within the ISP's network there's not liekly to be any routing information for that network, so even if someone within ISP A's network wanted to talk to this rogue address it'll get passed up through the gateways to a core router and tossed off into the internet according to its BGP routes. Even if there were routing information in ISP A's network for the bad netblock communication with the internet would be basically impossible.
Now, if you really want to screw up the internet you start advertising bad BGP prefixes.
Long story short, BGP is based on implicit trust, and any route advertised is generally accepted as correct by its peers in the BGP network. One notable example of this going horrifyingly wrong is when Pakistan decided it wanted to block Google some years back. Routes that were intended to be a BGP blackhole internal to the country were mistakenly advertised publicly, causing much of the world's Google traffic to be directed at Pakistan, taking the Google, and the PK ISP, offline.
Now, if you want to get scary...
With a suitable amount of network and computing horsepower someone could ostensibly poison BGP routes on a global scale, intercept and store traffic, and forward that traffic back out to its intended destination transparently. Something something NSA.
Note on the comments on the question: What @Sven means is that you shouldn't even use the phrase "Class A/B/C" as it refers to rigid, long-obsolete practices. If you want to be taken seriously by networking nerds you need to use CIDR notation, ie: '/8' instead of 'Class A'