Dumbledore planned his death by Snape killing him so that the Dark Lord will trust the Potions Master. In the Half-Blood Prince, The Ministry of Magic accepts that Voldemort is back. Hence, the former Minister of Magic, Cornelius Fudge was forced to resign his position and leaves the Ministry in disgrace. He was replaced by Rufus Scrimgeour. After the battle of the Astronomy tower, the two Ministry of Magic officials visit Dumbledore's funeral.
Now, the Ministry of Magic didn't bother to investigate the Death of the Headmaster. Particularly, where Rufus Scrimgeour is now in charge. Dumbledore even said that the new Minister is a "Man of action, doesn't underestimate Lord Voldemort". With that said, we could expect that the Minister would investigate Dumbledore's death. Of course, the Ministry doesn't know that Dumbledore planned to have Snape kill him. But looking at what Snape did, it causes shock among wizards and people in Hogwarts. Even the whole Order of the Phoenix and Hogwarts community thinks of him as a traitor. Therefore, the Ministry has a good reason to investigate the murder of Albus Dumbledore. They could've questioned Harry Potter since he was the only one who was there and is on the same side as Dumbledore. If they ask Harry, then the Minister could try to arrest Snape.
Take note: I'm referring to the Ministry of Magic prior to its control by Lord Voldemort. So therefore, I'm pertaining between the events of Half Blood Prince and Deathly Hallows.