First of all, foley is not just organic sound effects, you can actually make foley playing synthesizers and samplers as well, foley is the art of recording something in sync with the picture. Myself I almost exclusively use that for movement and footsteps, practically everything else is edited and designed from hard effects (my personal library).
With that said, good question! Myself I try as far as possible to use sound sources from organic and otherwise "worldly" stuff. "Worldly" here meaning of non-synthetic origin. For example, I have an old disc-echo in nowhere near mint condition. I will eventually fix it, I love these great old gadgets, but when I got it it had several broken parts and a really horrible magnetic coating, which gave an eerie pulsating sound with marvelous little discharges, sparks, and lots of audio artifacts! Though I could, and did, manipulate how the unit was operating, it still was nowhere near synthetic - it was a very living form of controlled chaos based in unintended feedback, unpredictable and unreliable distortion, corrosion, and heavy fluctuations (the motor was the first thing I replaced). Even had I wanted to I could never ever make the exact same sound ever again. I've used many of the sounds I made from that in many projects over the years without ever repeating myself.
This sound is, though all electronic (one must differ between electronic and synthetic, a Fender axe or Rhodes-piano is nothing without electricity, but seriously not synthetic), exactly like nature: It lives, breathes, and tho it might sound the same for 10 minutes in a row, it really doesn't. If I was to try and make the same kind of sound on a synth it would take a lot of work (I shit you not, though most of it is pure drone this sound is really insanely complicated when analyzed!), and it would most likely have close to no imperfections in the sound, meaning it might be hard to pull off as realistic.
That is however not always a problem! In my arsenal I have lots of synthesizers, drum-machines, vocoders and other electronic instruments ranging from early 70's (my Yamaha SY-1) to about 10 years ago (a Roland MC-505. Surprisingly useful). Everything from purely additive (making sounds by arranging sine-waves) to analogue subtractive synthesis. All of these synthesizers can be programmed to make personal sounds impossible to find in nature, but too awesome not to use in the mix. Take for example my Evolution EVS-1. This is an old 1U rack, at least 20 years old I'd say, based in several kinds of modulation-syntheses as well as additive. This little beast have absolutely no warmth nor power whatsoever, but I've made lots of absolutely stunning digital sounds with it! The KORG MS-20 is another synth I've made tremendous electronic- and robot-sounds with! As it's modular the possibilities are enormous. Not as fat as the PolySix or JUNO, but superior in versatility.
One of my absolutely best synths was really not a synth at all before I and Mister Soldering Iron invited it to a little surgery: my Commodore 64 Breadbox with ditto keyboard overlay. Forget everything about "8 Bit-Sound" here, the SID-chip is a mostly analogue synth-chip with additional wavetable noise. The sounds from this gets a little different from the other synthesizers, it has a wicked bite to it, but the fact that I can rout an external signal through it and automate the filters make it totally invaluable for work.
When it comes to more traditional synthesizers, I'd say they can add both bite and substance even to more realistic (not naturalistic) soundscapes as long as one uses them carefully and not drags too much attention to them and mix them with "worldly" sounds. A good example was one of the first movies I made. A werewolf jumps up to an ascending helicopter and beats the crap out of it. The first parts was easy, I made the whole sequence from sounds I recorded myself and based the helicopter on real helicopters I recorded outside my house combined with a dismembered former washing-machine. The descending was a lot trickier though. I wanted something really filmic, the sound of a really fritzy chopper-motor going completely haywire. I ended up with the combination of a screaming falling sound I made on my Kawai K1 II (wavetable additive) with sweetening from the washing-machine and a very big RC helicopter. For the final impact and plowing through the ground with the chopper I used rumbles I made with my JUNO-60 (Old Betsy. My very first synth!!) and percussive sweetening from an Ensoniq SQ-80, although all the essential sounds was recorded in and around my home.
However, it is very rare that I use synthesizers as prominently as I did in the helicopter-crash in the werewolf-short, especially since I actually began working with this. At least for sounds intended to sound natural and diegetic (ie: in the same worlds as the movie unlike music and stingers that's outside the movie-world). The rumble for example I do with a large sheet of metal nowadays, and what I don't have in sound effect elements I can easily record. The last few years I've made mostly horror, thrillers, action, and drama, so synth-sound just plain doesn't work at all. Possibly for occasional drones and stingers, but my organic sounds still worked better so far.
What does gain on synthesizers IMHO is SciFi! Not meaning robots and computers, that goes without saying, but for example forcefields (as you mentioned), hums, details, and special sound effects! With special sound effects I mean things like Star Trek beamings, Optimus Prime transforming, or even simple ones like the bass-fall from good ol' Obi-Wan being naughty with the tractor beam. As Tim said, nature is far more rich and complex than a synth can ever be, but if it's the right tool for the job, it's the right tool to use.
We all have different ways of dealing with things as well as different approaches and preferences - myself I quite frankly work very little with processing whatsoever. I try first and foremost to reach my goal with different kinds of microphone techniques, processed with only ordinary filtering, dynamic control, and reverb (often in that order), possibly combined with some nifty distortion, and as such synth-sounds rarely mix very well. But when using synths I still prefer to make my own patches, I seriously dislike using factory presets, and I only use hardware. Period.