Would the president telling the military to attack “cultural sites” be a lawful order?

Short answer, it is not a lawful order. The military already put out a statement saying we would not target cultural sites.

The answer to this is nuanced.
Protocol 1, Article 52 of the Geneva Conventions provides for the general protection of civilian objects, hindering attacks to military objectives. Article 52 states, "In so far as objects are concerned, military objectives are limited to those objects which by their nature, location, purpose or use make an effective contribution to military action and whose total or partial destruction, capture or neutralization, in the circumstances ruling at the time, offers a definite military advantage."

Any attack must be justified by military necessity: An attack or action must be intended to help in the military defeat of the enemy, it must be an attack on a military objective, and the harm caused to civilians or civilian property must be proportional and not "excessive in relation to the concrete and direct military advantage anticipated".
Some targets are clearly legitimate. These include all military personnel who are not Hors de combat. It also includes anyone who takes a direct part in military hostilities. It also include military equipment and bases and any buildings used as fortifications whether designed as such or used by the military ad hoc.

Civilian infrastructure such as, rail, road, ports, airports and telecommunications used for the transportation of military assets, or used by the military for electronic communications are all considered to be legitimate military targets.
Where it starts to get more nuanced is if the harm to civilians or civilian property is "excessive in relation to the concrete and direct military advantage anticipated". During the Second World War there was a song called a thing-ummy-bob which contain the lines "And it's the girl that makes the thing that holds the oil, that oils the ring that works the thing-ummy-bob, that's going to win the war". Whether such a girl is a legitimate target is an area that probably has to be decided on a case by case basis. However Protocol I suggests that if it is not clear, then the parties to the conflict should err on the side of caution as Article 52 states "In case of doubt whether an object which is normally dedicated to civilian purposes, such as a place of worship, a house, or other dwelling or a school, is being used to make an effective contribution to military action, it shall be presumed not to be so used".

In Civil Affairs, we produce what is called a "protected (or restricted) target list". It generally consists of things like hospitals (usually marked by protected symbols like the Red Crescent anyway), museums, schools, and key infrastructure. These are not things that we can not destroy, however. They are things that require additional approvals to target. The ROE, not the Law of Land Warfare, usually take precedence when engaging such targets.

In my opinion, "cultural" targets such as statues celebrating the Islamic Revolution or regime palaces may well be targeted in such a strike, as the objective would be to punish the regime, niot bring general misery to the people like strikes on power plants and communications infrastructure would.

I would encourage all of the armchair quarterbacks out there that are such learned experts on such things due to their prowess with google take the time to read the relevant information before rendering judgment.

IMO we should have not had gone down that rabbit hole.