In Ancient Rome, what type of government (and non-government) response might there be to large natural disasters?

Well, the treatment of house fires might give you an indicator about the sort of approach that would be taken.

The history of organized firefighting began in ancient Rome while under the rule of Augustus. Prior to that, there is evidence of fire-fighting machinery in use in Ancient Egypt, including a water pump invented by Ctesibius of Alexandria in the third century BC which was later improved upon in a design by Hero Of Alexandria in the first century BC.

The first Roman fire brigade of which we have any substantial history was created by Marcus Licinius Crassus. Marcus Licinius Crassus was born into a wealthy Roman family around the year 115 BC, and acquired an enormous fortune through (in the words of Plutarch) "fire and rapine." One of his most lucrative schemes took advantage of the fact that Rome had no fire department. Crassus filled this void by creating his own brigade—500 men strong—which rushed to burning buildings at the first cry of alarm. Upon arriving at the scene, however, the fire fighters did nothing while their employer bargained over the price of their services with the distressed property owner. If Crassus could not negotiate a satisfactory price, his men simply let the structure burn to the ground, after which he offered to purchase it for a fraction of its value. Emperor Nero took the basic idea from Crassus and then built on it to form the Vigiles in AD 60 to combat fires using bucket brigades and pumps, as well as poles, hooks and even ballistae to tear down buildings in advance of the flames. The Vigiles patrolled the streets of Rome to watch for fires and served as a police force. The later brigades consisted of hundreds of men, all ready for action. When there was a fire, the men would line up to the nearest water source and pass buckets hand in hand to the fire.

Rome suffered a number of serious fires, most notably the fire on 19 July AD 64 and eventually destroyed two thirds of Rome.

http://en.wikipedia.org/wiki/History_of_firefighting

Sorry to just copy paste Wikipedia on you, but they've made a decent write up of the whole thing.

I can also speak with a certain authority on historic fires in Bergen, Norway. Norway is and has been predominantly wood-builders, and Bergen is historically the biggest city, famous for having narrow streets (smug in Norwegian) and a lot of ships. From the ships you get tar and other kinds of flammable shit that they use to make the ships waterproof. So the city burnt down a lot, more than 20 major ones, if memory serves. The name of their football team is, not coincidentally, the Norwegian word for fire (brann), so it's a bit hard to google for a fact check. Anyway, when a major fire broke out, the whole city would work together: They had large pikes (like this but 20 meters long) that they'd get out and demolish the log-built houses creating zones of separation between the fire and the rest of the houses in the city. If this didn't happen, everyone's house burnt down. They would also put wet sail cloth on the sides of houses in an effort to stop the fire spending. This whole system was institutionalised in the 17th century, when they rebuilt with broad alleys in the city, that separated it into quadrants that would (had it worked) save those sectors that weren't on fire. It didn't work the first time, and the entire city burnt down in 1702. Since then, fires have been limited, tho, and you still see those streets in Bergen even though it's mostly stone constructions in the centre now.

There was a private fire brigade, organised by the German Hanseatic merchants (dry fish traders), who had a permanent trading post in the city (the wooden part of the city you see on post cards now) between 1360 and 1760. They had one horse-drawn pump wagon that held something like 100 litres of water, and were known for being quickly on the scene if a fire broke out. Of course, they lost their whole stock when the city burnt down.

Anyway, most relevant to your question, but what I know the least about: Not much (according to the book I'm basing most of this on, Bergen Brenner^1) is known about how people who were displaced by their homes burning down, but we assume they would go live in the nearby villages and go back into the city and help with rebuilding when they could.

Source: Worked a guide at the Hanseatic Museum in Bergen.

Edit: Also, expanding upon this, in the Christian middle ages, disasters were seen as acts of God, and the Church were the ones to provide (limited aid) to those affected.

^1 Not completely sure on the title, read it while bored at work. It might be Byen Brenner.