In the courts, all that mechanically changes is that dead people become witnesses.
I feel that they might not make for reliable witnesses.
(SPOILERS: Phoenix Wright: Ace Attorney)
In Phoenix Wright: Ace Attorney, the DL-6 case is a case in which a spirit medium is used to communicate with the dead, as to find out who the murderer is. Relevant excerpts:
On December 28th, 2001, Miles Edgeworth went with his father, Gregory Edgeworth, to the courthouse. Gregory Edgeworth had a trial against Manfred von Karma, a veteran prosecutor who hadn't lost a case in his long career. During the course of the trial Gregory managed to prove that Karma had tampered with evidence, giving him his first penalty in court.
After the proceedings, Gregory, Miles, and a court bailiff named Yanni Yogi left court and entered an elevator. Almost immediately afterwards there was an earthquake and the building lost power. The three remained trapped in the elevator for several hours with little oxygen. Yanni began to panic, and an argument broke out between him and Gregory. During the struggle that ensued Yanni's pistol came loose from his belt. Desperate to stop their fighting, young Miles took the gun and threw it at the pair. It fired once, and all three in the elevator finally lost consciousness.
The power then came on, and the elevator doors finally opened. Inside, Gregory, Miles, and Yanni were all unconscious, and lying next to them was Yanni's pistol. Already furious with Gregory for his failed reputation and under the shock of having just been shot, Karma snatched up the gun and shot Gregory in the heart, killing him instantly. He then left the scene.
The police investigated the murder, without finding any trace of the first bullet fired from Yanni's gun. But the second was a perfect match to Yanni's weapon, and he was charged with the murder. Even that, however, wasn't enough to convict him, and so the police turned to spiritualist Misty Fey (mother to Mia and Maya Fey). She summoned the ghost of Gregory Edgeworth to testify as to the events.
Whether or not Gregory really understood how he had died that afternoon, he was quick to place the blame on Yanni. Through Misty, he claimed that Yanni shot him during their struggle. Miles testified to the same. But Yanni's lawyer, Robert Hammond (employed by the Grossberg & Co. Law Offices at the time) coerced his client into pleading temporary insanity. By placing the blame on panic and oxygen deprivation Yanni was able to earn an acquittal and was set free.
Misty Fey's reputation as a spiritualist was ruined. Because the testimony she had "summoned" from Gregory Edgeworth failed to hold up in court, she was considered a fraud and a fake.
The reason Gregory here gives false testimony is because when he passed out, he was in a sealed elevator. No one else could get in. There were two other people in the room. His son, Miles, and some bailiff, Yanni. He had had an argument with Yanni. He passed out afterwards.
Surely he didn't kill himself. He had heard only one shot, and that was when his son threw the gun. He believes his son killed him, by accident. But he doesn't wish to indict his son. And Yanni was hostile to him anyway. So why not blame Yanni?
This is what I believe to be a relevant example for your world. People may mistake events that happen. I believe that people will have trouble recalling the moment of their death because it is a strange state to be in (dying, as the brain will start to conjure up things that aren't there due to lack of resources), and that their testimony simply isn't solid evidence. It's practically the same as a witness who was in a state of delirium.
The impact of this varies on how your medium/spirit summoning enters the world. If spirit mediums are recently new, expect courts to have suspicions of using the method. Expect people to express distrust.
If they're public knowledge and have been around for a long time, however, expect killings to be done differently. Murdering someone is something you have to do without exposing yourself to your victim. Murderers result from things like domestic violence or disputes gone bad are likely to be solved via testimony of the deceased, but assassinations or sudden deaths (hit-and-run?) are not.
It's not going to solve all cases. It's also not definitive evidence.
I also wonder how you're going to deal with perjury - what if someone who is dead lies?