IIRC WARP was only able to forward your origin IP to websites using Cloudflare. Then, as of Aug 2022, their FAQ[1] says your origin IP is hidden regardless of which website. Their IPs do reveal your geolocation though.
There was a bug[2] that revealed your IP to select websites; that seems to have been fixed by Nov 2022.
Disclaimer: I’m not knowledgeable enough to test every possible IP leak mechanism (like WebRTC), so I didn’t do that. I’m basically taking their word for it.
[1] https://developers.cloudflare.com/warp-client/known-issues-a...
[2] https://community.cloudflare.com/t/beware-cloudflare-warp-do...