No. When Claude, Cursor or Codex builds your site, it uses your own plan with them, so none of your BYQ AI tokens are spent.
There is a daily limit on how long your AI tool can keep your site open: 3 hours on Free, 12 on Pro and 24 on Ultra. Previews are free on every plan. To take the site live you need Pro or Ultra, or hosting for that site. Setup is in Build with your own AI tool.