I've heard many stories about monuments being deliberately preserved or spared -- perhaps the most famous being Hitler's decision to protect the enormous Canadian memorial at Vimy Ridge -- but I must confess that I know next to nothing about those that didn't make it through the war.
I only know of a few cases of purposeful memorial destruction. /u/DuxBelisarius talks more about that here. In general, as you note, many memorials were purposefully spared—often times, memorial sites are seen as shared sites of mourning, as I discuss here.
Of course, some memorial sites were damaged "unintentionally" during the Second World War as a result of bombings (that's probably pretty obvious, but still). I can speak most directly to sites of this sort in England, which is where I've studied them the most. Great War memorials can be found in almost every branch of the Church of England, and historic churches were certainly damaged and destroyed in the Second World War. In those, for example, the memorial sites were lost along with the rest.
The train car in which Germany signed the armistice to surrender to the Allies at the end of WWI was captured by the Germans from a museum when they overran France. At Hitler's order, it was then moved to the site of the original signing and used as the location where France signed their surrender to Germany. After this bit of symbolism, the train car was moved to Berlin as a trophy of war, but was eventually moved further east and ultimately destroyed by the SS towards the end of the war.
