For vowels, the chart actually shows a continuous space, with height, backness, and rounding corresponding to values we can measure (and synthesize): the formant positions.
For consonants, the chart isn't actually a perfect representation of how the mouth works. Some things, like the place of articulation of plosives and fricatives, are actually continuous: a human can make a palatal fricative, or an alveolar fricative, or a fricative anywhere in between, for example. And a computer can do the same.
But other things on the chart aren't continuous in the same way. Trills, for example, can only exist at certain places of articulation, because only certain parts of the mouth are flexible enough to trill. And there's not really anything acoustic in common between them, except that they're trill-like: there's not some clean measurable value that's highest for bilabial trills, in the middle for alveolar trills, and lowest for uvular trills, for instance. If there were, we could interpolate between the alveolar and uvular to make a velar trill, but unfortunately there's not.
Similarly, clicks at different places of articulation involve entirely different processes, so there's no way to "fill in the gaps" and figure out what impossible clicks (like the velar click) would sound like.
Other sounds are considered "impossible" because they're exactly the same as other sounds. For instance, a glottal trill is marked impossible. That's not because there can't be any vibration down in that part of the larynx, but because there's already a name for when the vocal folds vibrate and nothing else is articulated: we call it [ə] (in other words, the most central, mid, neutral vowel).
Now, there are some impossible sounds we can create using computers! For example, we can simulate a vowel that's higher and further to the front than [i], which is as high and front as the human mouth can go. Or we can go lower than [a], or farther back than [u], or any combination of these. I'm not sure if anyone's ever researched how people would perceive these impossible sounds: if they'd sound unnatural and unlike any other vowel, or if they'd just be perceived as [i], [a], [u]. That would be an interesting experiment to run.