The Claude (and ChatGPT) web UI supports incognito/temporary chats that are discarded when you close the tab. Click the ghost or dotted speech bubble icon in the upper right corner of the page.

I use this as my default mode so I don't clutter my chat history with random, one-off questions I ask. Unfortunately, there is no way to change your mind and save a productive incognito chat after you've started chatting.