For instance, if I have a circle and I want to make 1/16" breaks in the line in 3 or 4 places along the circle, how can I do this? I have tried the eraser tool, but it closes the line up around where I erased.
One way: Create small lines where you want the circle to break, then use the Virtual Segment Delete tool (Crop tool flyout) to remove the unwanted parts between the lines.
Break a circle.cdr
It can be helpful to explain the production situation - laser/wood/vinyl
Do you need equidistant breaks?
One commercial macro has a feature to put tabs where you like with a mouse click: See #2 here:
http://macromonster.com/product/shelbys-object-order-tab-creator-for-laser-cutters/