Talking to schizophrenics for advice how it sounds won't necessarily do you much good as we're not talking something like an LSD-trip or such here, we're talking a state that appears totally natural to the senses of the one suffering it, and this differs from person to person, as well as there are rarely dronings or similar sounds present. With rarely I mean that it does happen in severe cases, but they will be too personal to generalize.
I've known several people with schizophrenia through the years, and they all had different things to describe about their state, but I did also once in my early twenties accidentally put myself in a similar state by staying awake from somewhere between 8 and 9 days straight.
Though extreme sleep deprivation causes something more related to psychosis, movies often mix up psychosis with schizophrenia to the point that most people can't really tell the difference, and how extreme mental illness is depicted in film often crossover the two states.
There are two aspects of tackling this problem: Psychotic sensory - Which often seems to appear less natural to the one suffering from it, more dream-like, and narratively - An interpretive way of depicting what's on-screen but doesn't originate from reality, but being an un-diegetic invented narrative to the scene. Like woosches, music and such.
The dreamy part is easy - when you think about it, how did you feel and experience the last dream you had? How did you experience the sounds and colours? I'm working on a project right now that is supposed to feel like a 2h long dream, and there are much for anyone to learn by just analyzing their own dreams, but in addition to that, after nearly a week and a half of being awake I began to hear someone insulting me. Severely. The words was something I'd get banned permanently for from the entire interwebz and all he moderators collectively burning my computer to ashes....still, there was never any words. I understood them completely, but they wasn't there. They had no direction, but I followed them successfully to my coffee-cup in the kitchen. I was wide awake, but I was dreaming. And like in dreams, there was only the flat here and now, no details, no dimension, no background, only the apartment that existed but never was. The best way to depict something like this is to off everything in moderation, like slowing down things, replacing them with other components that doesn't quite jive, and, most prominently, use a heavy unhealthy dose of EQ. A dreamy sound have substance but have no core. The gut-ripping scene in The Cell is a marvelous example, but regretfully this is also really really hard to get right. Removing these frequencies from the human voice for example might very well make it disappear completely from the mix on no-so-great sound-systems, killing the effect completely.
Those were a lot of words, I know, but I'm afraid there is only so much you can do by using these techniques, using it correctly might very well turn the scene into something more than what it should be and must therefore be used in moderation. Narrative sound design though is not used to recreate reality, but the feeling. It's not used to tell WHAT the character fells, but HOW it feels. Here - used in the same way as music - heavy dissonance, dronings, unreal effects, the works, can be used to depict the feeling of inner chaos, maybe insecurity, and a warped mind. The lead word here is "Inharmonic". Also "Don't drown everything else (unless you have a reason to) and don't over-do it, this isn't diegetic mind you...". Forget the usual cliche about low bass-tones - bass can be used for calming as well as distress, what makes it threatening is mimicking snarls and growls, which isn't based in low bass but in the lower mid, and also rich in sometimes slightly dissonant overtones. Another good component is rhythmic sounds without rhythm, played in the sweet-spot pace were they're not too slow to keep consistency, and not so fast that they make a random rain-pattern and becomes more of a carpet of sound. Yet another extremely useful component is human voices, mainly whispers, and animal sounds, all heavily processed to remove them from diegesis and not attracting too much attention to themselves.
One thing is perhaps the most important of all though - never forget that a personal state of mental distress and disarray is a heavily personal state - make the character live >IN< the world. There is no front or back from this, there is only here and in the middle of it all, and there is no order, only chaos. Have something everywhere over the entire soundfield, have extreme pannings, have interaction and indifference at the same time...but do not have substance nor presence - this isn't real nor even what the sufferer actually experiences, only feels.