You should evolve. Thanks for listening :-)
Actually, I'll expand on that.
You're right that C99 has been around for quite a while. You should (in my opinion) be using that standard for anything other than legacy code (where you just fix bugs rather than add new features). It's probably not worth it for legacy code but you should (as with all business decisions) do your own cost/benefit analysis.
I'm already ensuring my new code is compatible with C1x - while I'm not using any of the new features yet, I try to make sure it won't break.
As to what code to look out for, the authors of the standards take backward compatibility very seriously. Their job was not ever to design a new language, it was to codify existing practices.
The phase they're in at the moment allows them some more latitude in updating the language but they still follow the Hippocratic oath in terms of their output: "first of all, do no harm".
Generally, if your code is broken with a new standard, the compiler is forced to tell you. So simply compiling your code base will be an excellent start. However, if you read the C99 rationale document, you'll see the phrase "quiet change" appear - this is what you need to watch out for.
These are behavioral changes in the compiler that you don't need to be informed about and may be the source of much angst and gnashing of teeth if your application starts acting strange. Don't worry about the "quiet change in c89" bits - if they were a problerm, you would have already been bitten by them.
That document, by the way, is an excellent read to understand why the actual standard says what it says.