There are many artists performing with single pickup guitars, playing various genres. Some even remove two pickups from HSS guitar and leave only bridge one. When it beneficial to do this?
Update: This might be worth checking
Electric guitars have a long sustain (and can be used for acoustic feedback for that reason) because the string does not lose significant amounts of acoustic energy. If you have good bridge and nut, the main loss of energy might be through the magnetic fields of the pickups, through magnetization losses of the string even if the pickup is switched off (if it's on, losses might be more complex). So it's conceivable that reducing the number of pickups will lead to longer sustain and easier acoustic feedback if that's part of your playing style.
"conceivable" means that if people do quantifiable experiments which show a noticeable difference, I won't cross this off as fantasy. But without such experiments, I remain rather skeptical.
Of course what may very well be the case that with heavy play some of the pickups might be a problem mechanically, being hit by the string.
If you know what sound you require, and it is from one pickup (ie if you always keep your pickup selector in the same place) then there is really no need to have other pickups. Every pickup can be a source of interference and noise, so removing them reduces the potential for noise.
Some also do it for the look, and in the past some have definitely done it because a pickup broke and they didn't get around to fixing it :-)
If you want the flexibility a multi-pickup configuration gives you, don't do this.
Even pickups that are not wired in to the circuit at all can have an influence on the vibration of metal strings due to the magnets used in the pickups. However, this is only likely to be noticeable when the pickup's height is such that it is very close to the strings. This video(start at about 1 minute) suggests that the 'warble' that this creates is much more noticeable when it's the pickup being listened to that is too close to the strings, but if you had two pickups in close proximity (with one of them being disconnected but too high) it's also likely that the effect could be heard easily.
(It should be noted that just because the effect is easily audible, it doesn't necessarily mean that it's all due to the effect on the string's vibration - it is also likely also because the added proximity alters the relative balance between the modes of vibration picked up by the pickup.)
Since electric guitar strings are made of metal and magnets inside the pickups are attracted to metal strings, they will have an effect on each other.
The more pickups you have in a guitar, the more the magnets in the pickups are affecting the free vibration of the strings.
When pickups are properly height-adjusted this "pull" is minuscule, though I'm sure measurable in some way (it is hard to find tests out there in the real-world of this, though, like Tubes, it just "sounds" difference to "trained ears" - there's a perceived sustain increase). The only real evidence (something you can test yourself) is when pickups get too close to the strings. Amplified you'll notice the sustain is cut off a bit, and there's a "warble" (as mentioned in other answers) to the sound where the magnetic pull is greatest on the string and weird science takes over.
So generally, the fewer pickups in the guitar, the less magnetic pull and more free-string vibration you have. Also, the closer the pickups are to the strings, the more magnetic pull they have (since they're stronger and closer). The more magnetic pull, the harder it is for the string to vibrate freely without restriction.
There are other small factors with more or less pickups; the most relative probably being that the less-routed guitar body has more mass to affect the sustain of said strings. This is really only apparent though in top-mounted guitars. Generally, modern Strat-style guitars come with H-S-H routes (or even boat routes!) under the pickups for universal fits regardless of whether the guitar only has a single humbucker or is truely H-S-H.
Another relative factor has more to do with the tone-circuit. Most people who use just one humbucker generally have no tone pot in the circuit. They often just have a volume pot going straight to the output jack. This simplification of the circuit leads to a no-load configuration that will increase the output a bit, as well as preserve some high-end otherwise not perceivable in a traditional volume-tone circuit. ...Judging by how common this particular configuration is to single-humbucker guitars (and you asking specifically about people who remove all but one humbucker), I thought I'd include this as well.
EDIT: Check out this link on Music: Stack Exchange. It goes more into the physics of this.
Since pickups have permanent magnets, it doesn't matter if there switched out of the circuit, they still have the same influence on the vibrating properties of the string unless you physically remove then from the guitar. The input load into the guitar amp is very high, so it probably won't make much difference if it's in or out of the circuit.
If you want to know if the magnetic field from a disconnected pickup alters or dampens the string vibration - try this test, take a pickup unit that has been removed from a guitar, and move it close to the strings just like a normal PU except from the top. Play an open string slowly with the pickup close to the strings, and then move it away, and see if there is a difference in the sound.
I would like to add to these answers another thing. Well two actually. 1. Having open space below the strings can sound better than to close them in with pups. Giving them room to breath. Try listening to your sound unplugged closely. Then lower the pick up as low as can go, and listen again while still unplugged. You can here a difference. More so if the guitar lacks body sound. And is why they sometimes sound better lowered. And not because of the commonly thought reason. Guitars vary in this. So being close may get better sound in one way. And worse in the other. Finding the happy medium is the trick. 2. Some folks like me pick rather deeply. And the pups get in the way of the pick! Having the pick strike the pups is rather rough at times. Especially if one was only used to playing the acoustic which has nothing close to the strings. Some can't get used to playing around the pups. Which is why I would take out the ones I was not using, or could get by without.