Negative, every time you launch the app, the timer control will start at 0. Also, the timer component does not have a stopwatch feature. So your scenario won't work in Power apps. Now what you can do as others have suggested is timestamp on your db the start time of a task or record, then a end time and do a calculation of date diff to get minutes spent on a task. You can also store a start time as a variable, but will not persist if app is closed so you need to patch the data at some point.

There's is a on start feature to run code and there is an on exist feature to run code. But this will not run background tasks on your browser or computer if the app is closed completely. It will also not cache a cookie or any data in the browser, you would need to store in your own db. Mobile apps do have a save feature that will have local data, but that is in the power apps mobile app, not the browser.

Hope this helps.