I made a screenshot of the different classes that you can use for styling a default tab hover.
The class you can use and see if you get a border is the following:
.tundra .dijitTabHover {
border-style: none;
}
For Checked Tabs you need to use the following:
.tundra .dijitTabCheckedHover {
border-style: none;
}
Hope this helps.
If you've tried border:none and it doesn't work, you've probably applied it on the wrong css class. Try to identify the right css class which is applying this border and then you can remove it.