I'll take the occasion to mention Brandon Sanderson Mistborn series. Its practically the main theme of the second book.
Anyway, to make it short: any empire needs some sort of power infrastructure to work. It serves no good being an evil emperor if you don't have anyone who goes around, making sure every peasant stays in check.
Most evil empires are set in a fantasy/medieval word, so we can mainly talk about nobles, army lieutennants and priests, but this could really be true for any administrative figure. An army of tiny, zealous bureaucrats can work as well as a real army. After all your empire cannot sustain itself on pure evil: you will need someone to collect taxes for you, spread your orders, cull the locals, enforce law ... tl,dr: a lot of people, organized in some sort of gerarchical structures, are needed to make an empire function.
Now, when the good guys take down the evil boss, they'll have to deal with this state of things. Let's assume that the evil army isn't a problem anymore, because it's being conquered already, and rule them out (it's kind of a strong assumption, since there may still be rogue evil-battalions around, or experienced generals gone into hiding). The good guys have to choose how to rule - turns out, as you can imagine, that is not easy.
I'll also assume, from now on, that the best option is to keep the empire united and intact and try to avoid total chaos. A central government means order, and the good guys probably wouldn't want to be remembered as the ones who saved the land to let if fall into mayhem.
Sit on top of the previous infrastructure
One option is to convince all the government apparatus that there is a new boss in town. This could seem a quite viable option at the start, since the good guys have the upper hand with the military power.
This way poses several hidden and not-so-hidden threats.
First of all, we may as well assume that most of the administrative higher-ups are evil, or corrupt, or have tainted themselves with horrible atrocities. Probably it will be hard for the good guys (from now on, the GGs) accept to work with them.
But this is true for lesser bureaucrats as well. Who knows, after all, how much corruption is spread into the empire organization?
Probably most people who worked with the evil emperor felt that they were "just doing their job at the best of their capacity", but as it turns out, those people tends to be cruelly efficient when led by a convincing evil intent (If you haven't read it: Eichmann in Jerusalem - Wiki).
Some can be convinced to work with the GGs, some will retain faith to the previous emperor and try to silent undermine the establishment. Given months, the collection of taxes or food sources may get more and more crunky.
Keep this in mind, because it can be crucial.
Kick the castle down
Seeing how the previous option fares, the GGs may want to kill/exile/fire everyone who ever worked in the previous administration and replace them with people of trust and goodwil. And that's great! I mean, I'd have too my best friends leading far, important regions of my empire, rather than some possibly murderous stranger.
The point here is that if the evil emperor or his staff cared even a bit about efficiency (and someone, if an empire is able to sustain its weight, has to care about it) it means that the previous administration knew the work well, or at least, well enough to do it. The GGs are maybe good at winning the war, but may not have the sligthest idea on how to do accounting, how to collect and store taxes and supplies, prevent famine, disease spread, overpopulation in the main capital ... and whatever else.
It might take time to build a new system that works. This means a long period of instability and reduced tax income at least, total catastrophe at worst. Probably the GGs are on the good side of a lot of people, exspecially towards those categories oppressed by the evil ruler, but even those people will stop being supportive without a job, a shelter with decent conditions, and food in their bellies.
Also, large masses of people are prone to have a short memory. "Hey, the last emperor was kinda bad, but at least was able to produce bread for all!" regardless than this was true or not.
So, it's probably good to keep your army around and try to mix between the two solutions - giving most power to people of trust but trying to learn all the possible from old employees before, if ever, replacing them.
How to deal with nobility 101
If your empire has nobles, you probably have to win them on the GGs side. Maybe there are some nobles on the GGs army (that was all good to overthrow the evil emperor, but now they'll want something in return...). Nobles tend to be quite proud, stingy, and good at scheming, thought - not exactly the people you want around. As it goes, they could be a major source of instability - even if not openly hostile towards the GGs.
In every empire in human history, there was the struggle between the central authority and the aristocracy / local authorities. This is going to stay, no matter what. Even the most good willed noble will try to get something for himself - now, imagine what the least good-willed can do.
Nobles conspiring to assassinate, or start a rebellion, are a clear example of this. They could right up take their lands, fortify them, and proclaim their region a separate reign. Either they will do this right away, or at a most propitious moment. Hostile nobles can work with reluctant bureocracy, zealous priests, and - even! - other nobles.
Even if they work with you, their are still regional authorities, and giving them too much power will be detrimental to the empire - assuming you want to keep the empire on, of course.
Yea, I know. Most evil empires are so evil and so big that there is literally nothing beside them. If the previous emperor didn't conquer all that was there to conquer, tho, the GGs will have to deal with their neighbours, they be either other empires/kingdoms/coalitions of city states or "whatever sort of dweller of the wild regions". Now, the second kind of neighbour can be an hassle to your border region, but should be in check (unless they manage to unite all the tribes in one enormous raiding party, of course).
The real treat are those neighbour kingdoms. Maybe they hold a grudge against the empire, maybe they just want more land for themselves; the point is, the best moment to wage war to someone is when he isn't able to respond. Probably the GGs will have to wear energies and resources to mantain order and internal affairs, so, it will be difficult for them to resist an assalt from a foreign nation. Better still if this happens at the same time with a noble revolt in another part of the empire.
Much and more:
I kinda spent half an hour or more on this answer, and you'll be tired of reading my rants. Is there more? Oh, yes. Surely someone else will bring up religious matters (more so if the evil emperor was lead of a cult), economics, and who knows what else. Hope I did provide some good insights, tho.
So, quick recap:
Does a new government rise up from the ashes of the empire led by said good guys? Yea, but it's a ton of work.
Or would the land just fall in to chaos now that none is around to govern it? Could definetly happen, regardless of the GGs intentions. Lead an empire is no easy task; in real life, empires often didn't survive their emperors (Alexander the great is a great example of this). Chaos can be temporary, tho. Some sort of order should emerge from it, after much bloodshed.
Would it be possible for them to face the same hardships as their enemies, like dealing with rebels wanting to rid themselves of their own style of government? Absolutely. But before dealing with rebels, they have to deal with all the administrative duties! Common routine, yes, but the larger the empire, the more difficult it gets.
What could possibly happen after the empire falls? Chaos, as you stated it; more war, as I explained earlier, between different factions; or a longish period of instability and insecurities after wich a new empire eventually rises.