You cannot use Time Tracking without Budget Management being turned on, since the rate system is part of Time Tracking even if you choose not to use it. However, you can use profiles to completely hide all financial fields from the end users even if Budget Management is turned on. Here is a tutorial video on using profiles: https://success.clarizen.com/hc/en-us/articles/220336187
Making sure users do not have Financial Permissions will also hide financial fields and functionality.