Colour theory has a lot to do with how the brain processes the signals from the retina, as well as the physics of how light is detected in the eyes. But broadly speaking, the additive and subtractive properties of colour result from the physics of light and its interaction with pigments, so if we were tetrachromatic we would experience them similarly. The main difference is that a tetrochromat would experience four primary colours rather than three. This would change some aspects of colour theory (I guess you'd have something more like a "colour sphere" than the colour wheel, for example) but the basics would be the same.
Imagining we had perfect RGB screen technology, capable of reproducing all the colours within our visible spectrum range (400nm—700nm). Could you, if you had tetrachromatic vision, use the same RGB screen and still see all the colours in your "visible spectrum" of the same range?
Such a screen is actually kind of impossible. There are colours that we trichromats can see that cannot be reproduced by an RGB screen. The range of colours a three-colour screen can produce are only a subset of the colours we can see. This is due to the way our cone cells respond to more than one light frequency at once. It's a fundamental thing, it's not just due to deficiencies in the screen. But the subset of colours that an RGB screen can produce is a pretty big one, so we generally don't notice.
But aside from that point, no, a tetrochromat could not use a three-colour screen and still see most of the colours in its visible range. There would simply be a primary colour missing. It would be like trying to reproduce all the colours a trichromat can see using only two light frequencies. If you use red and cyan for example, then you can get red, white, cyan and black, but you can't get blue or green.
Could you use this perfect RGB screen technology if, while trichromatic, your cone cells were "tuned" to different frequencies (but still covered the identical "visible light" range)?!
No, you couldn't, for similar reasons. Most likely, it would still cover a large proportion of the colours you can perceive, but there would certainly be some colours you can see that it wouldn't be able to produce.