BeoVision 10 has two fans actually, they are referred to in the Service Manual and can be adjusted (at what temperature they turn on, and what's the speed of the fans at minimum and maximum temperature). They are located in the lower half of the TV, where all the electronics and speakers are housed.
Luckily I've never heard even a hiss from BeoVision 10, even in complete silence. BeoVision 7, which I had a couple of years ago, was also totally silent.
In the unlucky case of fan noise, perhaps they can be adjusted on any BeoVision by B&O service, making them run more silently. There is a recent post on fan noise regarding BeoVision 8, explaining this in more detail.
KMA
Current setup: BeoVision 10-46 (grey speaker cover, AR, motorized stand) with Apple TV 2 (FireCore), Sony BDP-S780, Mac Mini, BeoLab 11 (silver), Beo5, BeoSound 8 (red speaker covers). Accessories: A8 Earphones, wine bottle coasters.
B&O product history, in chronological order since 1990, after the onset of the treaded BeoVirus (I tend to upgrade/change my setup "infrequently"): BeoSystem 2500 (with blue speaker covers), BeoLink 5000, BeoSystem 7000 (complete; silver/black), BeoLink 7000, RedLine 60.2, BeoVox Penta, BeoVision MX4000 (black, motorized floor stand), BeoCord VX5000 (black), BeoSystem 4500 (complete), BeoCenter 9500, BeoLab 8000, BeoLab 6000, BeoVision Avant (original, 28" AR, VHS, green), BeoCenter 2300, BeoVision 3-32 (grey speaker frame, AR, motorised cabinet), DVD1 (grey), BeoCord V8000 (grey), Beo4, BeoSound Ouverture (w/ floor cabinet stand), BeoVision Avant RF (grey, 32", AR, VHS), BeoSound 9000, BeoCenter AV5 (blue), BeoVision 1 (yellow, motorized floor stand), Beo1, BeoSound Century (yellow), BeoCenter 1 (blue, AR, motorized floor stand), BeoSound 1 (silver, floor stand), BeoVision Avant RF DVD (grey, 32", AR), BeoVision 7-32 MkI (AR, motorized floor stand), BeoLab 3 (black), BeoSound 2, BeoVision 10-40 (grey speaker cover, AR, motorized stand).