Great question. This is the exact reason why you will always hear people say:
“Never edit the core code”
Often, editing the “core code” is the quickest path to creating the result that you’re after. But it’s an incredibly terrible practice to follow - And often you’ll find inexperienced/budget developers do this for a quick buck. (This sometimes justifies the reason for paying higher hourly rates for the experience).
The truth is, that when you edit core code it can result in a couple of issues:
- The changes you made are overwritten when you update Magento
- Magento is unable to be upgraded due to code conflicts
When I first started to use Magento 1.4, my predecessors simple “winged it”. Little did I know that I would suffer from the aforementioned issues that would end up resulting in me having to rebuild Magento 1.7 because I couldn’t upgrade or patch it. And if you can’t upgrade or patch it, then you’re opening the door to further issues.
Consider this…
Let’s assume for a moment that your core code has changed. And that the latest upgrade replaces these files. So, you think “Oh, I’ll just upload my files again”. Well, consider this:
Why were those files updated in the upgrade? Did it fix a bug or security patch? Is it enabling a new feature? If you simply end up uploading your old changes again to make it work then you’re also deleting the latest updates to those files.
The right way
The reason for doing things the more complicated/expensive/longer way by using overrides, file-extensions and child_themes is that they still work (within reason) regardless of an upgrade. Thus making it more secure and future-proof.
Anyway…
To answer your question, if your custom changes were all developed using best practice (such as overrides and child themes) then the answer is most likely going to be “No, you don’t need to do anything”. Otherwise, see above!
Reading back, this reply probably came across quite heavy and passionate. I didn’t think my answer would be this long. But when it comes to experienced Magento users/developers, then we’ve probably already learned this lesson the hard way which makes us feel so strongly about it 
Anyone should feel like they can ask any question here. If you couldn’t find the answer on here to begin with then that means the topic wasn’t mentioned before or I did a poor job of making it visible and that’s on me. Ask whatever you need to ask 
and in case they have not , they are liable to do it properly because they did tell me that they were not going to touch the core files and they follow best practices.