Only if you hide the icon as well. If that's acceptable, you can use the Systray customisation dialog to change the relevant properties for Eraser to 'hide icon and notifications'. Personally, I wouldn't do this, as I like to know when a task is finished (and many other programs now use the notifications in any case). At least the notifications intrude into the workflow a lot less than the Eraser 5 modal progress box.
The problem is I need the taskbar to appear only when I pass the cursor down there. It's an option that I have been using forever. I am so used to the extra acreage. When the Eraser message causes the taskbar to popup, I need to be always clicking on the message to get rid of it and the taskbar so I can use the slide control at the bottom of my Photoshop canvas. Well, it's no big deal, really, because it's not like I am running Eraser night and day.
I sympathise, as I have had the same issue. Unfortunately, the problem is built into the design of the taskbar. The idea is the notification is presented as a 'speech bubble' emerging from the tray icon; for that to happen, the icon, and therefore the taskbar on which it sits, have to be displayed ...
Since writing the above, I have realised that there is a much better way to resolve the problem.
Eraser does not show the popups if the Eraser program is open. So if you have, say, Photoshop open full screen, you can use the tray icon to open Eraser. You can still use the context menu to start Eraser tasks if you prefer. Once the task is started, you can click on the Photoshop window to make it sit on top of the Eraser window. The Eraser task will complete without popups.
Note that running the more disk intensive Eraser tasks (erasing free space or large amounts of data) is probably best not done when another program, particularly one as resource intensive as Photoshop, is running at the same time.