Slate's Explainer answers, "Either the sheriff or the medical examiner"; but I don't think that's quite right.
A suicide note is a tangible item that was owned by the person who committed suicide; the property rights in it, as with all other property, go to the person's heirs under the will or under intestate succession rules (if the person died without a will). The government may well have considerable rights to hold on to the tangible item for quite a while, because it's evidence in the investigation of the person's death. But that doesn't mean the government now "owns" the note, only that the law allows it to keep temporary custody of the tangible item.
The same is true of the suicide note as a copyrightable work. Any note that's longer than a few words is protected by copyright, even if there's no copyright notice; that work is originally owned by the author, but when the author dies it goes to the author's heirs. The government's temporary custody of the note may physically block others from copying the copyrighted work, but the government doesn't own the copyrighted work.
As the Explainer points out, the government may also have -- depending on state law -- have the duty to place the contents of the note in the public record. I expect that such state laws do not violate federal copyright law, because such placement would be "fair use," despite the general principle that fair use is rarely available as to unpublished works. And fair use may also allow the media and others to further reprint what they see in the public record.
But none of this makes the government the "owner" of the copyright in the suicide note, because the government lacks a quintessential right of an owner: The legal right to exclude others from using the property. If, for instance, the suicide's heir reads the suicide note before the police show up, the heir may then publish the text of the note, and the government can't stop him: They don't actually own the copyright. If a third party publishes the text of the note, perhaps that might infringe the copyright in the note (unless that too would be fair use, which it well might be) -- but it would be the heirs, not the government, who would decide whether to sue over the infringement, because it is the heirs who own the copyright.
Now, as I said, it might well be -- especially if the note is placed in the public record and can then be copied freely from there (presumably on a fair use theory) -- that everyone would then be able to publish the note without copyright liability. But that just means that the note effectively wouldn't be owned by anyone: It would effectively stop being property at all, much like the works of Shakespeare stopped being property. (I say "effectively" because, unlike with Shakespeare's works, the note would still be technically covered by copyright law, but by hypothesis that formal protection would have no effective bite because of the broad fair use rights.)
So, the suicide note as tangible item would remain owned by the heirs, though the government would have temporary custody (as the body of the Explainer item discloses). The suicide note as copyrighted work would either be owned by the heirs -- subject potentially to some considerable rights on the government's part to place the note in the public record -- or would be effectively no-one's property, if the fair use rights are so broad that anyone would be free to copy it. The copyrighted work would never become the government's property in the sense of the government's having the right to stop others from using it.
Finally, this highlights a broader point: One often hears that the options are whether one entity owns the property or another entity owners the property, for instance "Do consumers own their personal information or do merchants?" But there's also often a third option -- no-one owns the property any more, and it's free for everyone to use. That's generally the case, for instance, as to facts (such as information about certain transactions, which the consumer, the merchant, and anyone else who learns about it are all free to disclose). And it may end up being the case for suicide notes that end up in the public record.