Are there any Wii games that come with firmware upgrades on the game disc which require you to install them before playing the game? If this is the case how do people with flashed Wii's get around this?
First: Your Wii is not an XBox360 or PS3, dude. There's no "flashing".
Now, to your question. In the end, it all comes down to the IOS, and you can forget about the Systemmenu, until now, its version doesn't have any effect on games. Users who use only a cIOS to load their games, either from USB or the Disc, just need to update their cIOS (probably), because all games will only run on this one custom IOS, and all other IOS won't be used. Users with ModChips, Darkcorp or SNEEK will (sometimes/probably) need to update their IOS, or install a new one, because the game will run on the IOS it requests with these. Darkcorp-Users will just wait for a new version, while SNEEK-Users also probably need to wait for an Update for their App.
How to do this: Savest and easiest way is to use DOP-Mii WiiBrew-Edition, with the latest HBC and IOS58. You could also use Mod-Mii/NUS-Downloader to get a .wad and install it with a WAD-Manager. If the IOS doesn't exist yet on the Ninty-Server (happend with IOS58, but was then later put on the Server), there are several ways to extract the new WAD (for example, with WiiXPlorer) from the Disc, or search in the Internet for the WAD and then install again with the WAD-Manager.
Oh, and about the games: It's not very common, but from time to time, it happens. Examples were: NSMBW, Your Shape, Need for Speed: Undercover, etc., etc. If it is the case that a game requires an updated/new IOS, you'll hear it here at GBATemp.
Oh, and forgot to say: You can use Priiloader/Startpatch to block Updates, from the Disc and from the Update-Button.
Slow your role, DUDE. Wii updates are FLASHED to the NAND, so yes there is flashing. There is no need for an SD card and there is no HDD... so where did you think it went?
The correct answer to the OP's question is... it depends on what firmware you currently have installed!
If you have all the latest IOSs, then no. OTherwise... maybe. But as Blue-K was trying to explain, with a hacked Wii you can simply install all of the latest IOS files without changing your system menu version. For your system menu, I recommend staying at/upgrading to 4.2.
Shure to the NAND, dah, I'm not that stupid. But then, every Wii would be "flashed", and in the end, his question wouldn't make any sense (I'm pretty shure he was going for the flashed DVD-Drive from the XBox360, which simply doesn't exist on the Wii, but you probably also know that, right?).
Though, if there's a next time, I'll add "of a DVD-Drive", to make it more clear, ok?