So I know this is so many years later and Halo 3 is almost dead on the 360 at this point in time, but from the testing results I have gathered is that whether you install or don’t install Halo 3 to the HDD can either have quick load times or slow down load times, but all in all it has no effect on gameplay whatsoever which is great news, let me explain… even though Bungie tells you not to install the game to the HDD I’ve noticed that the code that they are using to copy the disc based multiplayer maps to the utility partition on the HDD is not 100% efficient when it comes to load times either, by going in a custom games lobby I have cycled through the disc based multiplayer maps and the DLC multiplayer maps and have noticed that sometimes the disc based multiplayer maps and the DLC multiplayer maps load quickly but not always, however when I cycle through the maps again to see if they still load in a quick state they don’t always load in a quick state like they are supposed to considering the disc based multiplayer maps are being cached to the HDD and the DLC multiplayer maps are always on the HDD, the loading bar will take its time loading to 20% and after 20% the loading begins to speed up, not sure why this happens but what it’s telling me is that it’s not 100% efficient or guaranteed to load the disc based multiplayer maps or the DLC multiplayer maps quickly all the time, after finding these results I went ahead and installed Halo 3 and tried to see how much of a difference it made, using the same testing I did without the disc installed I noticed that at first most of the maps would take it’s time to load like before trying to reach 20% and then after 20% it would speed up, and some maps would load quickly like they are supposed to, but after loading them once sometimes the maps would remain loading quickly if you tried to load them again and some would slow down again, not sure why this happens since Bungie said installing Halo 3 would result in longer load times but from what I’m seeing that is only happening 50% of the time, basically the answer that I have found out of all of this is that whether you don’t install or whether you do install Halo 3 the loading mechanic is not 100% efficient either way, which means it would affect matchmaking load times whether you installed or didn’t install Halo 3 but not always, so my answer to you is yes I would install Halo 3 to gain the benefits of always having a clean install of the disc, and also so your system runs quiet and so it doesn’t put so much wear and tear on the optical disk drive by spinning the disc all the time which is creating more heat inside your Xbox 360 making the fan work harder to dissipate heat, hope this helps for all of you, 
The waer and tear on the optical dics drive killed my old Xbox 360. That is why i got myself an external hard drive for my new one. Since then i always instill my games.