Hi,
Storing the task attachments in a different DB may not be possible. However you should be able to customize the task details view and decide whether you want to show/remove the task attachments panel.
However note that these changes will be applicable via the Appspaces only. To customize the task details go to Appspace > WHN demo if it exists. If the WHN Demo appspace does not exist, create a new appspace and add the gadget - Business Console Common > Workstream. Save the appspace.
The Workstream should show all the task types. Hover over the task type you want to make modifications. You should see an downwarrd arrow option, click on it. You should see the option Customize Task Details page. Click on it and then edit to remove the task attachments panel gadget. Save it. The task details page should now no longer show the task attachments panel.
Note that these changes will not apply to the out of box task inbox in the business console. You will need to create a new Appspace and then add the Gadget - Business Console Tasks > Task Inbox.