In terms of bombing, cultural sites generally weren't specifically targeted. Partly this was practicality - even in ideal conditions hitting a single building was difficult, at night when much strategic bombing happened even finding the right city was a challenge. Precision raids required considerable planning and preparation so wouldn't be wasted on a target that had little material impact on the war; destruction of a landmark may have affected morale, but was less of a concern than the deaths, injuries, homelessness, loss of utilities, etc caused by large-scale bombing. Both sides also generally attempted to present their own bombing as careful attacks on military targets in contrast to 'terror bombing' by the other, for their own populations and (particularly prior to the US entering hostilities) neutral countries. Damage to famous buildings could even have the opposite effect, not demoralising but stoking anger and determination to fight. Buckingham Palace was not specifically targeted by the German Air Force during The Blitz but was hit several times, most notably on September 13th 1940. The King and Queen were present and not taking shelter at the time; the Queen wrote to her mother-in-law about the experience:

"It all happened so quickly that we had only time to look foolishly at each other when the scream hurtled past us and exploded with a tremendous crash in the quadrangle," she wrote.

While her "knees trembled a little bit", she was "so pleased with the behaviour of our servants", some of whom were injured as one bomb crashed through a glass roof and another pulverised the palace chapel.

Despite the popular images of British stoicism during The Blitz there was disquiet as lower class tightly packed housing around the docks was disproportionately affected by raids and East End residents felt 'the toffs' weren't being hit; as Harold Nicolson wrote in his diary: "Everybody is worried about the feeling in the East End, where there is much bitterness. It is said that even the King and Queen were booed the other day when they visited the destroyed areas (...) if only the Germans had had the sense not to bomb west of London Bridge there might have been a revolution in this country. As it is, they have smashed about Bond Street and Park Lane and readjusted the balance." The propaganda opportunity of the attack on the Palace and the King's heroic survival was obvious, journalists were rapidly invited to come and inspect the bomb damage and as the Queen famously said, "I'm glad. It makes me feel I can look the East End in the face."

Into 1942 there was the 'Baedeker Blitz', so called when von Stumm of the German Foreign Office announced "we shall go all out to bomb every building in Britain marked with three stars in the Baedeker guide". That would have been tricky as a single star (or asterisk) was used in the guidebook as a mark of commendation, with a few particularly notable places receiving two stars. Historic cities (Exeter, Norwich, York, and Canterbury) were attacked, but not really specific buildings within them - on the first raid of April 23rd only one of 40 aircraft actually hit Exeter at all, the rest scattered bombs across Devon, Cornwall and Somerset. Even Goebbels thought von Stumm's announcement was counterproductive, writing in his diary "I censured this in the sharpest terms and took measures for preventing the repetition of such folly".

Though not specifically singled out for cultural reasons historic buildings were often damaged accidentally, sometimes deliberately for military reasons. Perhaps the most notorious example was the monastery of Monte Cassino, destroyed by heavy bombing as it was believed German forces were occupying it, though it turned out they were not.

There were at least some efforts to prevent damage to cultural sites, largely under the auspices of the Monuments, Fine Arts, and Archives (MFAA) unit of the Allied Military Government for Occupied Territories, sometimes known as the Monuments Men (as per the 2014 George Clooney film) or the Venus Fixers. Rome was a particularly thorny issue with the presence of the Vatican. Roosevelt, facing elections with substantial numbers of Italian-American and other Catholic voters, was initially strongly against any bombing; Churchill, without such considerations, had no objections. The Archbishop of Canterbury wrote to the Secretary of State for Air asking for Rome to be spared but the reply was that military objectives must be hit regardless of their proximity to old buildings. Charles Portal, Chief of the Air Staff, was in favour, though he noted that some of his crews were reluctant to bomb targets such as Rome, Florence or Venice; Anthony Eden and the Foreign Office were strongly opposed.

The Foreign Office view held sway until Operation Husky, the invasion of Sicily. Eisenhower's headquarters identified two rail marshalling yards in Rome important for moving German reinforcements and sought permission to bomb them. The yards were three miles from the Vatican so the attack was sanctioned providing the crews were carefully briefed on the importance of accuracy.

Leaflets were dropped on July 18th 1943 warning of the attack, which followed on the 19th, led by Jimmy Doolittle (of the eponymous raid on Tokyo) who offered Catholic pilots the opportunity to step down, though none did. Crew were briefed in detail with the strictest instruction to avoid four Major Basilicas including the Basilica of St. Peter in the Vatican, but without a detailed report from the MFAA (it had only been formed the previous month) intelligence on sensitive sites was not complete and the Basilica of San Lorenzo, in close proximity to one of the marshalling yards, was not marked on maps to be avoided. It was hit during the raid and seriously damaged, with frescos and mosaics destroyed.

With the MFAA established there were more rigorous efforts to catalogue significant sites to be marked on maps for the briefing of aircrew. As attacks on Italy stepped up with the invasion of the mainland, cities were divided into three categories: the most important (Rome, Florence, Venice and Torcello) that could only be bombed under specific instruction; a second category including Ravenna, Assisi and Pavia that were not regarded as of major military importance but could be bombed if necessary; and a third including Pisa, Siena and Bologna that, despite architectural importance, contained military objectives and could therefore be bombed freely.

Rome was largely spared, other than the Basilica of San Lorenzo, as German forces withdrew rather than strongly defending it. Florence was bombed in March 1944 (with authorisation from Churchill) with even more careful planning than the Rome attack, avoiding 58 important monuments, though the Germans destroyed bridges as they retreated leaving only the Ponte Vecchio. Padua, on the other hand, was bombed on the same day with less care, destroying the Church of the Eremitani, and other cities including Pisa and Naples suffered heavy damage as they were fought over.

Overall the MFAA supplied information to prevent sites from suffering damage where possible, and did what they could to repair and protect sites once the fighting had passed, but faced constant opposition from those for whom military expediency was the prime, if not only, consideration, and the latter generally held sway.

For a quick overview [How the Monuments Men Saved Italy’s Treasures] (https://www.smithsonianmag.com/history/how-monuments-men-saved-italys-treasures-180948005/) is a nice piece about the MFAA from the Smithsonian; if you'd like to drill into original documents Fold3 has a large collection from the Roberts Commission - Protection of Historical Monuments including handbooks, lists of monuments, and maps that were prepared at the time.