In this question it was established that the US President can be a dual citizen as nothing in the Constitution forbids it. This makes me wonder if there are currently any heads of state who are also citizens of a second country? Obviously excluding honorary citizenship or situations where the second citizenship has been renounced before taking the office.
The current president of Somalia, Mohamed Abdullahi Mohamed, is a dual Somali-US citizen.
Andrej Babiš, the winner of the 2017 election in Czechia holds dual Czech and Slovak citizenship. He became the Prime Minister and Head of Government, after coalition negotiations in December 2017.
The Wikipedia article on Alberto Fujimori (President of Peru in the 90s) states that he has both Japanese and Peruvian Nationality, and this had been obtained by his parents, who were both Japanese citizens. I'm slightly doubtful, as I understood that Japan doesn't allow for dual nationality, but the situation may have been different in the past, or there may be special arrangements for Peruvian Japanese.
Yes, the head of state of Vatican is expected to keep his original citizenship. At the time of writing this, Francis I is a citizen of Argentine. I do not know what would happen if a citizen of a country that automatically strips the citizenship if someone acquires another citizenship (such as Slovakia or Japan) is elected.
The President of France is always the Co-Prince of Andorra and The British monarach is a citizen of her commonwealth.
In case of Sweden, the head of state, the monarch can't be a citizen of another country.
Our citizenship laws does now allow dual citizenship ie a swede can now apply for citizenship in the US and keep his/her swedish (i note this because we had a case of swedish OS-hockey player who had to leave the troupe after the first three games in Nagano in 1998.)
The permission of dual-citizenship also includes politician including members of Riksdagen and the head-of-government including all the ministers.
This is muddled, partly because the law and the material considering transfer of head of state (? successionsordningen) only says that person eligible as next head of state (monarch) must be a swedish citizen and being a member of the swedish church.
The citizenship laws changed in 2001 and 2003, well after the material about successionsordningen (which is from 1977-1978 with other changes since that.)