There appear to have been a lot of building of monuments in the interwar period and from what I could see not too much has been talked about what happened to these memorials immediately before and during the war.
What were the attempts to save these memorials, if any? Were there attempts, say, by Nazis to destroy Allied memorials? If so, why? And what of the historical battlefields themselves, Ypres, Passchendaele, etc?
I can't speak to all of the memorials, but during the war some were destroyed in France and Belgium by the Germans during the war. The Mont St. Quentin memorial, depicting an Australian soldier bayonetting a German eagle was destroyed in 1940; the Vimy Ridge Memorial, by contrast, had a detachment of SS men to guard it during the war. The statue of Ferdinand Foch at Bouchavesnes was left in place, but the Germans destroyed much of the scenery around it, in an act of spite to the man that had, essentially, won WWI (the Germans still lost, again; Foch 2, Germans nil).
I do not know of evidence that points to widespread destruction of memorials. As has been mentioned, the Nazi army did destroy a memorial specifically depicting violence against a German eagle. I've also seen reports of WWI battlefield and trench markers being trampled by moving troops (however, those reports were indirect, the result of a student interviewing a Belgian man, and I'm somewhat hesitant to trust them at all). But that's about all I know of. Certainly many memorial sites were left untouched, and those that were damaged were often casualties of troop movements or bombings rather than specific targets of violence.
War memorials were, and are, often shared sites of mourning between nations -- even nations that were opposed in wartime. In the case of the Second World War, for example, the community of Coventry Cathedral's Cross of Nails was in contact with German religious figures even before the war ended, and the cross is visible in churches on both sides of the war, around the world. I am not well informed on the subject of memorial preservation and destruction specifically in wartime, but it would not surprise me to learn that memorials were avoided and protected by both sides. As has already been mentioned, I know that Hitler himself took personal interest in the preservation of a Canadian memorial at Vimy Ridge. I'm not saying that was standard procedure, but I do know that so many Great War memorials were left untouched following the Second World War that it was standard procedure to save money on a new memorial by adding the names of fallen WWII soldiers to an existing WWI site.