Audition can work on the audio track of a video just as it can on any other audio recording. When Audition does noise removal (more precisely reduction), it does it by working on an audio file on your hard drive, not live as you play back or make the recording. So you need to have a way to isolate the audio track from the video recording, either by using Audition to rip the audio track digitally or make a real-time analog recording of the video's audio track as it is played through your sound card, to get it on your hard drive, so Audition can work on it. Then when you're done, if you still want the audio track synched back up with the video, you'd have to do that as well. Adobe Premiere or the right piece of video gear can do that part, but not Audition, AFAIK.
What's hard for many people to understand about digital noise reduction, and this is true for any audio software including Audition that can do it, is that there is usually no way to completely isolate and remove any one sound from a mixed-down recording assuming it has similar frequency response properties to other sounds on the recording. This is why ryclark used the "unbake a cake" metaphor: you can't remove just the sugar from a baked cake without affecting other ingredients. You can remove a vocal if the recording was made in isolated tracks and you can delete the vocal track, but you can't remove the sound of the singer's voice once she's mixed in with a band, because her voice shares many sound characteristics with other instruments. It sounds different to you, but it looks the same when subjected to digital frequency analysis. Same thing when you try to remove voices from a video. So hum and hiss are much easier to reduce than voices, trumpets, the drum kit, the background noise you hear in a restaurant, etc., all of which span a wide range of frequency response.
To do noise reduction with Audition, you must first capture at least a brief couple of seconds of just the sound you want removed. Then the software scans the audio file and dampens way down the frequency response of the captured sound. It will reduce but not completely remove that sound and all similar sounds. If you have no way to isolate the sound you want removed so you sample only that, noise reduction is generally ineffective.
There are two parts to your question.
1. Is it possible to use Adobe Audition to edit the audio track in a video? Yes, with some execptions. Exceptions include an inability to play the video through your computer's speakers, or issues with your PC and Digital Rights Management if your hardware enforces it. But if your PC has analog inputs, and you hook up say, a video camera to those analog inputs and play the video through the speakers, and your PC allows analog recording, Audition can record the audio track of the video. If your hardware allows ripping of digital audio from a video, and you rip the file, Audition can edit that file.
2. If I use Audition's Noise Reduction to remove vocals and background noise from the audio track of a video, will I be happy with the way the background music in the video sounds? No, almost certainly not. Audio editing software can do a good job in many cases of greatly reducing a sound that operates across a narrow frequency range, such as tape hiss or hum. It does not do a good job removing vocals or background noises in a mixed-down audio track so that all that is left is the music. I will be very surprised if you like the results, trying this. This is not due to limitations of Audition per se but of digital audio editing software generally. You are asking it to do something it cannot do at all well. The only way to do what you want is to have access to pre-mixed tracks so you can punch out the tracks you don't like. Again, the unbake the cake metaphor: you cannot remove one ingredient from a baked cake... right? I think you're going to have to come at your problem a different way, other than audio editing software, for acceptable results.
If you want sugar extracted from a bake cake, you'd be better off going to buy sugar in a grocery store.
If you want the music off a video's audio track, you get a recording of the soundtrack, or if that doesn't work, get access to the pre-mix master tapes used by the film's producer (where the music is on an isolated track), or you find and hire musicians who can recreate it. Occasionally, a commercial video is released where you can select under languages to hear just the music soundtrack. I think that's happened with a few Danny Elfman scored films, but I know that's rare. I just have no way to un-mix a mixed-down sound recording as you need to do here.
Those who have told you that what you're asking for can't be done--by ANY software--are correct. The "unbake a cake" analogy is a good one. Once an audio track is mixed together, you'll never be able to completely isolate one element from another--there's too much overlap. If you think about the frequency range of the human voice, you'll realise that it overlaps with most common musical instruments. Since software can't know whether that 400Hz note is from a voice or a guitar (just an example), it'll either cut out everything at 400Hz or leave it all in.
Having said that, depending on how the track has been made and mixed together, Audition is probably one of the best pieces of software out there at playing with this sort of thing. The results can be anything from amazingly good to amazingly bad depending on the source material though.
Some of the built in Audition tools to have a look at and play with would be:
Effects/Stereo Imagery/Centre Channel extractor which lets you boost or try to remove parts of the audio track. However, the effectiveness will depend entirely on how the mix you're playing with was recorded and too much use can really destroy the bits you want to keep.
Restoration/Noise reduction: can be effective on any really constant background noise but, again, over use can mess up your music.
EQ can let you cut certain specific frequencies and boost others. However, as said, the human voice spans a wide range of frequencies shared with musical instruments.
Spectral View tools let you actually draw around certain parts of the audio and either just delete them or fill the hole with other sounds--extremely powerful but very time consuming and takes practice to get good at doing this.
(I'm not going into the details of how to use the individual tools--each subject would need several pages of instructions. The Audition manual or some of the online tutorials are a better bet once you know what to look for.)
So, to sum up, no software in existence can do what you want at a simple press of a button. However, if you are will to accept that the results will never be perfect and will often be dire, Audition has one of the best range of tools to ATTEMPT this of any DAW on the market--and certainly the very best range at its price bracket. If you've head instrumental versions of music used in the background of TV shows, that's because, at a professional level, companies just buy access to the unmixed music tracks. However, only trying and playing will let you see how close (or not close) Audition can get to what you're after.