Google Tasks – Hide completed tasks, don’t delete them

You could try CompanionLink, which is supposed to sync Outlook to Google, and vice versa.

You need to use a third-party service called Google Tasks Porter. I never used it, so I don't know how it works, but Cnet has a nice article that describes the procedure.

1. Point your browser to Google Tasks Porter.
2. The first time you use it, you need to give it permission to access your Google account, which takes several clicks. (If you use multiple Google accounts with multiple Tasks lists, you will have to approve each one individually.)
3. To start exporting, you'll need to take a snapshot. Click the big button on the left and you should shortly see a list of snapshots, which at this point should contain just one.
4. You can export to HTML, iCal, Outlook, and Remember the Milk formats by clicking the appropriate link beneath the snapshot info.
5. Each task manager has its own import tool, but all you need to do is point it to the file you just created. It may take some time, especially if you've got a long list, but it should work out well.
6. To start importing items into Google Tasks, just click the big Import Tasks button on the main screen.
7. From the import screen, you'll need to create a name for the new list, and choose a file to import from either Outlook or iCal. Of course, you will need to use the export tool for either of those programs to create the file Google Tasks Porter will use. Once you start, it should just take a short time for your new Google Tasks list to be created.
8. That's it! This is a pretty simple, straightforward tool that should make life easier for you if you want to give a new task manager a whirl without having to manually recreate your to-do list.