It's not entirely clear.
The examples mentioned thus far (in the question and answers) are possible cases of self-homing patronuses, but there is no definitive evidence in those cases. There is another instance where there is perhaps stronger evidence that patronuses can be self-homing. In Chapter Nine of Deathly Hallows we find the following dialogue:
“I say we find a quiet place to Disapparate and head for the countryside. Once we’re there, we could send a message to the Order.”
“Can you do that talking Patronus thing, then?” asked Ron.
“I’ve been practicing and I think so,” said Hermione.
“Well, as long as it doesn’t get them into trouble, though they might’ve been arrested already. God, that’s revolting,” Ron added after one sip of the foamy, grayish coffee.
Here they plan on sending a message to unspecified Order members at an unspecified location, and no one mentions anything about needing to know where the Order member(s) would be. At this point, they knew that The Burrow had just been overrun by Death Eaters, so that would probably not be the location, and they already knew that Grimmauld Place wasn't safe. That doesn't leave them many options of known locations in the first place.
Furthermore, Ron's comment about getting them into trouble would be a lot more significant if they didn't know where they were sending the messages to. If there was a possibility that the patronus would deliver the message in front of someone undesirable then there would be a real concern about getting someone in trouble; if, however, the patronus would be going to a specified secret location (such as one of the Order members' houses) then there would be less concern.
From this case it seems likely that the patronuses are not necessarily sent to a specific location, but to a specific person (or people).
On the other hand, in Chapter Eight of Half-Blood Prince we find the following:
"I meant Hagrid to get the message," said Tonks, frowning.
"Hagrid was late for the start-of-term feast, just like Potter here, so I took it instead.
This seems to indicate that the message was not sent to Hagrid, but was sent to the place where Tonks thought that Hagrid would be. If patronuses were self-homing the message could have gone to Hagrid wherever he was, instead of being intercepted by Snape.
Additionally, if patronuses are not self-homing we can easily explain various instances of wizards failing to use this method of communication. For instance in Chapter of Order of the Phoenix:
Dumbledore heaved a great sigh and then said,
“Alastor Moody, Nymphadora Tonks, Kingsley
Shacklebolt, and Remus Lupin were at headquarters
when he made contact. All agreed to go to your aid at
once. Professor Snape requested that Sirius remain
behind, as he needed somebody to remain at
headquarters to tell me what had happened, for I was
due there at any moment. In the meantime he,
Professor Snape, intended to search the forest for
“But Sirius did not wish to remain behind while the
others went to search for you. He delegated to
Kreacher the task of telling me what had happened.
And so it was that when I arrived in Grimmauld Place
shortly after they had all left for the Ministry, it was
the elf who told me — laughing fit to burst — where
Sirius had gone.”
Here we see that Snape did not send messages to Order members wherever they might be. He sent one message to Grimmauld Place, as that is the location he was aware of. Even more telling is that he had to assign Sirius to wait for Dumbledore, and Sirius in turn had to delegate that task to Kreacher. If a patronus could have found Dumbledore on its own, surely one of them would have sent him a message immediately!
If patronuses are indeed not self-homing, we would have to assume that Arthur had some other way of knowing that they were at Grimmauld Place. It might have been a lucky guess, or he might have sent a patronus to every possible location he could think of.