The answer definitely isn't universal, many cultural and historical centers were completely destroyed or looted during the war, while others were intentionally avoided or only lightly targeted when it was possible to do so. Going though each and every instance would be exhausting, so I'll touch on some of the major instances of attempts to protect historically significant sites, the most notable attempts being during the campaign in Italy.

/u/Bigglesworth_ answered a related question not long ago on the bombing of Rome here, where US and British factions argued back and forth for and against bombing cities like Rome, Florence, and Venice. A sort of compromise was reached where Rome in particular was only lightly bombed, and air crews were lectured on the increased importance of accuracy for these missions (especially since an important rail junction marked for bombing was only three miles away from the Vatican). Strategic bombing was also limited as much as possible for areas like Siena, Florence, Venice, and Milan, where mainly the outskirts and strictly military-related areas were targeted, and efforts were made to avoid damage to the historic centers to these cities.

Of course not every site of significance could be protected, most notable being the monastery of Monte Cassino, which was completely destroyed by the Allies as Axis forces occupied the building. Other areas that suffered extensive damage from bombings include Palmero, Messina, Treviso, and Naples, which was by far the hardest hit of Italy's major cities with an estimated 6,000 civilians dead and the destruction of the Basilica of Santa Chiara.

Aside from armed forces making intentional efforts to minimize damage to historic centers during instances that allowed it, nations also made the effort to protect their cultural centers by declaring them to be Open Cities. Cities that publicly announced that no effort to defend the city will take place in order to avoid unnecessary damage were classified as such. Major cities such as Krakow in Poland, Brussels, Paris, Athens, Rome, and Florence declared themselves as open cities when it became obvious that advancing armies were approaching the city.

If anyone would like to chime in on examples from other fronts, please feel free to do so. I do know many historic cities such as Kyoto were intentionally avoided, but I don't have enough source material on me to fully go into other regions, I would love to hear about such instances to avoid damage to cultural heritage in other areas of the world.

As for further reading, check out Forgotten Blitzes, France and Italy under Allied Air Attack, 1940-1945 by Claudia Baldoli and Andrew Knapp.