It's canon that Lupin and Sirius embrace like brothers at the end of Prisoner of Azkaban, and that they were best friends for years, and that Lupin is loyal to a fault. Couldn't Sirius have used Lupin's fireplace to visit Harry in the Gryffindor common room, rather than breaking into another wizard's house at 1:00 am to use their fireplace? Couldn't Lupin (Or Dumbledore, for that matter) have sent Sirius food in the cave so Sirius wouldn't have had to eat rats? Is Sirius just being melodramatic about the rats? Where is Lupin?
Lupin wasn't financially well off, he could not afford to have Sirius around.
The stranger was wearing an extremely shabby set of wizard’s robes that had been darned in several places. He looked ill and exhausted. Though quite young, his light brown hair was flecked with gray.
Harry Potter and the Prisoner of Azkaban, Chapter 5, The Dementor
He even mentions how hard it is for a werewolf to find a job.
I have been shunned all my adult life, unable to find paid work because of what I am.
Harry Potter and the Prisoner of Azkaban, Chapter 18, Moony, Wormtail, Padfoot, and Prongs
Werewolf were not trusted or liked.
“And I’m not a very popular dinner guest with most of the community,” said Lupin. “It’s an occupational hazard of being a werewolf.”
Harry Potter and the Order of the Phoenix, Chapter 5, The Order of the Phoenix
Sirius staying with Lupin is a major risk for both of them. People who don't like Lupin (or werewolf's in general) could file fake complains to cause troubles for Lupin, which would expose Sirius.
Secondly, I don't think Sirius wanted to cause any trouble to Lupin. Considering all the troubles Lupin was already into.
Lupin's whereabouts are never really addressed in the Goblet of Fire.
After Voldemort's return, when Dumbledore begins activating the Order of the Phoenix, Lupin is mentioned briefly.
Sirius, I need you to set off at once. You are to alert Remus Lupin, Arabella Figg, Mundungus Fletcher - the old crowd. Lie low at Lupin's for a while; I will contact you there.
Harry Potter and the Goblet of Fire, Chapter 36 - The Parting of the Ways
That's not much to go on. Presumably it would be unwise to hide an escaped convict with a known werewolf.
After he was sacked at the end of Book 3, I don't know that we have any evidence to support the idea that Lupin had a home, much less a fireplace, during Book 4. I don't recall him turning up again until he's living with the Order in Book 5.
As for asking Dumbledore for food, does Sirius really seem like the kind of guy to be begging people to send him food. His godson is one thing, the almighty Albus Dumbledore is another.