Bit of a mixed bag:

1. Intentional destruction of historical sites was not especially common, largely for the wasted effort angle (explosives are finite, time is a resource, spending either just to spite someone is not efficient). There are exceptions to this of course (like the Germans blowing up the rail car world war one ended in). Graffiti and vandalism however could be quite common. In Allied liberated portions of Europe at least, if either of these became unseemly MPs or other guards might be stationed to protect such places.

2. With that said given the lack of precision in weapons, and the sheer scale and intensity of combat, special protections only carried as far as practical (or no commander was going to sacrifice men or the mission to preserve a historical building, no matter how important it was). For a good representative photo, look up Nuremberg 1945 in google images. That's basically what a German city looks like after it's been hit with strategic bombing for some months, and then fought over in ground combat for about 100 hours. There might not have been the intentional destruction of historical locations, but if there was a fight, there was going to be damage. Sometimes a lot of it. Some consideration was taken to protecting sites when possible if it served a larger value, Kyoto in Japan was spared an atomic bombing because of it's cultural value (and Henry Stimson's insistence on the importance of that value!), the imperial palace in Tokyo was not intentionally targeted (this however was more to avoid killing the Emperor, as his death was seen as something that would make ending the war much harder), but it was still hit by errant firebombs anyway.

3. As far as precautions, it's another mixed bag. One technique was declaring an open city, or "surrendering" as a town, basically letting the invader know there was either no enemy contained within, or there would be no resistance. This of course was contingent on the willingness of the defender to part with the population center, and how much the attacker trusted the defender to behave. As examples, Manila in the Philippines was declared an open city in 1942 to prevent it from being heavily damaged by urban fighting (which proved only to be a stay of execution, the IJN opted not to return the favor when the Americans returned and the city was virtually destroyed). Many German towns towards the end of the war also aggressively surrendered if they could at the first sign of US or Commonwealth forces to be spared highly destructive assaults (generally the Allies honored these surrenders, although there are cases of retaliation when Allied forces were attacked post-surrender). Also as mentioned in terms of the Imperial Palace or Kyoto, some areas were placed off limits for targeting, however it stands to reason had those areas become more militarily important that protection would be removed (i.e. there might have been measures to avoid bombing the Coliseum in Rome, but had the Italians/Germans set up a weapons factory in the building, it would have almost certainly made its way onto a target list).