There's not really an "erase" tool (unless something's changed recently) as everything is defined as paths, etc.
However, you can sort of trick an erase in a number of ways:
1) Use an object the same colour as the background, and make sure it is "above" the other elements in the scene. You're essentially making a mask. This is similar to your white brush technique in GIMP.
2) Use an object and subtract from your other object. If you have multiple objects, copy and paste the object you want to subtract and then subtract them from all objects in the same spot - this is a little tricker but you get an element that truly has a piece missing. You may have to deal with paths and removing a section between nodes, and if you've got a fill colour it might not show properly, etc...
Ultimately, it might be easier to erase what you want to erase in GIMP after export, but if your diagrams are simple enough, a mask like in 1 will probably suit you well.
Hope that helps!