Since the building of the great works of Rome, the city has gone through massive shifts in its own purpose and significance, as well as going through periods of destitution, foreign occupation and times when the inhabitants cared about other things more than the vast expense of stopping its old buildings crumble.

By 500AD, the old capital of the Roman Empire had fallen on hard times. It had been sacked twice in the previous century, was no longer home to an Emperor - the remnants of the Western Roman Empire being ruled from Ravenna by an Ostrogothic leader, had little food supply following the barbarian expansions in North Africa and its population had collapsed from over 650 000 in the middle of the fifth century to little over 100 000 by the dawn of the sixth. The people of Rome had little reason to carry out massive public works of maintenance and for this reason the great works, such as the Colosseum and Circus Maximus, began to crumble.

It would appear that Romans had little nostalgia for their past, as the Colosseum was actually used for accommodation and workshops from the sixth century onwards. The decorative marble that once covered the Colosseum was also stolen and used on other buildings that were deemed to be more important.

The rise of Christianity meant more of an emphasis on church building than maintaining great works that were symbols of Rome's pagan past. The Pantheon provides an example of this new priority, changing from a disused pagan temple to the Church of Saint Mary and the Martyrs in the seventh century. Its continued maintenance was because it was still important to the people of Rome.

A useful parallel that comes to mind is the Parthenon in Athens, Greece. It had been one of the most important sites for Hellenic identity and culture, then became a church in the fifth century. It finally became a gunpowder store for the Ottoman occupants of the city. Practical use was prioritised.

The two examples demonstrate the same conclusion. Just because, a city has been inhabited constantly since a great work has been built, does not mean that the city's priorities and ideals have stayed the same, nor that they have had the funding to maintain their great works.

EDIT - My thanks to Azand for making me aware that the Parthenon was likely not a pagan centre of worship, and to Cleopatra_Jones for pointing out my lack of clarity concerning the political status of the Western Empire.

Source: Peter LLewellyn, Rome in the Dark Ages, 1996.

The Colosseum is an interesting example. While we find it fascinating due to its age and (now) frozen in time decay, people a thousand years ago would just consider it an old sports arena. Much of the appeal now for tourists is the fact that its ruined. Its visually striking.

Consider the Astrodome in Houston, Texas. The building is significant historically. It was one of the most groundbreaking feats of engineering in the last century.

Its now sitting empty, and even though locals find it historically significant, no one wants to pay to keep it up for historical reasons. Its far too big and cost intensive.

Its going to be torn down.

here is favourite piece of mine from 'urbs roma, a source book of classical texts on the city and its monuments' (ronald r. dudley 1967):

the emperors leo and majorian to aemilianus, city prefect. while we govern the state, we are anxious to correct obnoxious practices which have long been allowed to deface the appearance of the venerable city. for it is obvious that public buildings, wherein consists the whole beauty of the roman state, are on all sides being destroyed by the most deplorable connivance of the city administration. while the requisite materials for public buildings are being collected, the noble constructions of antiquity are torn down, and a great desecration is committed to allow a trivial repair. this has given rise to the practice that allows even individuals building private houses to filch and transport without hesitation, and with the indulgence of the city magistrates, what they require from public buildings, although all that contributes to the magnificence of the city should be kept in good repair by the zeal of the citizens.

this text is from a novella (or law) from A.D. 458 - as the destruction of ancient rome was happening. donald r. dudley goes on in his comments:

the document is one of an endless and largely unsuccessful series of attempts by the emperors to preserve rome from her most relentless enemies - her own citizens, who from the fourth century onward worked at her destruction with a sustained assiduity far more effective than the occasional onslaughts of the barbarians.

You might check out Peter Llewellyn's Rome in the Dark Ages. The volume is one of the few works on the topic.

Regarding the question at hand, I can offer a couple of observations. First, by the sixth century, there was little secular civic administration to do anything about preserving monuments such as the Flavian Amphitheater. By this point the Church had began to take over some of that work. Indeed, by this point, the only people still allowed to wear the ceremonial dress of the old senatorial class were priests.

The population of the city had been diminished greatly by this point, but was still unable to sustain itself without some sort of grain dole. Gregory the Great, whose family owned extensive lands in Campania and Sicily donated much of his property to the Church for the purpose of feeding the city. During the principate, the sites where the grain dole was distributed had temples located where this work was done. Gregory had churches built to accomplish this purpose.

I think this last anecdote is illustrative of the answer to your question. As the Church began to adopt administrative tasks, they were prone to constructing their own buildings. Indeed, as another contributor has commented, the Church had very little interest in maintaining buildings from the pagan past that could not be converted to ecclesiastical use.

Then of course, there was a lack of funds to do any of this. Beyond the financial crises previously mentioned, Belisarius's wars would have made it very difficult for anyone still participating in commerce by that point to continue their work, dying up external revenue streams that could have been used for this purpose. Additionally, any monies still in circulation would have been directed towards these wars.

when did societies become interested in preserving their edit, archeological, pasts? everywhere I go, ruins were re-purposed and no one cared until "recently," it seems.

As others have said it was the structures were destroyed by the repurposing of dressed stones from the original structure for the living inhabitants.

It is not unique to Rome. For example, the Great Pyramids of Giza were originally covered in white limestone and was polished to a smooth surface. What you see today is just the structural stone that remains. For centuries leaders, and conquerors removed the limestone casing stones and used them to build mosques and other civil infrastructure throughout Egypt.