The first thing is, don't make it totally gone at any point. Listen to the audio of medical dramas like Grey's Anatomy; the sound never really leaves the soundtrack completely. Fade it down and up, but not completely out.
Second, use viewpoint changes in the camera editing to adjust the level of the sound; take the opportunities when the camera moves further away from the sound source to reduce the sound level in a more natural way, then it's up to you whether to being the level back up when the viewpoint returns to put the monitor back in frame. You'll to pay attention to your stereo/surround image as well to make it believable; with the sound in center field while the camera has the monitor in view, when the camera moves right the sound moves left and v.v.
Third, you can usually get away with fading "background" sounds down slightly as a character starts speaking (or is just about to); we've become accustomed enough to the compressed sound of modern mastered audio that this doesn't sound jarring. Some of this really is from compression inherent in mastering and in broadcast transmission, but you can use a touch of it yourself in media formats like movies that are typically less compressed.
When the audience needs to remember the sound is there, or pay attention to a change in that sound, you simply bring it back up to prominence in the mix.