To break the site. But you're right that a lot fewer people will probably want to break it than to scrape it, and that stuff like CAPTCHAs is usually more about the "scraping" case. So basically a mistake on my part.
The sender does not have a direct communication channel with the receiver. They send the payment to a hop they are connected to (they have a channel with) and it gets routed to the receiver. The first hop would already drop an invalid payment. If they spam them with more invalid payments, all that would happen is that their connection to the Lightning Network would get lost as their channel partners would disconnect from them. The receiver would not receive a single network packet in the whole process.