No, I'm afraid I don't know of a simple way of doing this. You are a little out of my normal working path there.
If I had to do it, I'd probably import the audio into Audacity and mute sections within it. You'll have nothing but silence in those sections however.
Once you have the sound track as you want it, you save then remux the video with the new audio stream.
Now all that sounds simple but is not. Audacity, with the ffmpeg and Lame libraries installed, will open the avi directly and import only the audio so that you can work with it. I'd advise stretching out the time line a bit (using the zoom tool), so that it is easy to find the exact spots you need to mute. Then using the selection tool, click at the start of the section, drag the cursor and highlight the area you need muted. Click on Generate-->Silence. Do this for every section where you want to mute the audio. Save your project as an mp3 file, which should work well in an avi container.
Next step is to open the original avi file in AviDemux, go to Audio-->Select Track, then deselect the audio stream. Save as an avi, using Copy for both video and audio, and using the AVI muxer. This will give you the video only.
Now, remux the video and audio streams into an avi using AVI-Mux GUI.
There have to be better ways of doing all this, however, so you might want to get a free trial edition of Sony Movie Studio Platinum 12 and figure it out within that program. Or the equivalent programs from Pinnacle or Corel.