Is there a way to change the folder where a service is published in ArcGIS Server Manager?

No. There is no out-of-the-box method to change the folder from ArcGIS Server Manager once the service is published. When a service is published to a folder, it has a fixed REST URL and the URL cannot be changed. If the service is removed from the folder, it cannot be used by any application or web map utilizing the service. Therefore, it is best practice to organize the folder structure before publishing the service.

An alternative is to re-publish the service as a new service to a new folder. However, this method requires the new service to be added to all the applications or web maps that were using the previous service.