Please state the name and full version number of the Story Format you are using, as answers can be vary for each one. Because you didn't state the version (or even the series 1.x or 2.x) I needed to test your issue using both v1.2.3 and v2.0.1 of Harlowe.
Are you sure you are using the Play option and not the Test option to run your story?
I used the Play option of Twine v2.1.3 and no "Debug Mode" option appeared while running the story, although that option does appear if the Test option is used instead.
P.S. You can used the Developer Tools of your web-browser to alter what is currently being shown on any web-page, how you actually do this depends on the brand of your web-browser and the brand of your operating system.
On Windows using Chrome it is as simple as placing your mouse cursor over the visual item you want to effect, then use the Right Mouse Button to open the context menu and selecting the Inspect option. Within the Elements panel place your mouse cursor over the <tw-debugger> element, then use RMB to open then context menu and select the Hide element option.