Not in my case. I only give Terminal and iTerm “Developer Tools” permission. Cursor shows up under “Full Disk Access” with a toggle so it may have requested the permission at some point, but I have it on disable; I don’t see why it needs to reach out of directories I actively open. (And VSCode which I used for years doesn’t even show up there.)

Disclaimer: I’m not sure whether Cursor inherits iTerm’s permissions when launched from CLI. The TCC system is pretty mysterious to me in general.