You could program a MIDI device to play pitches this way when triggered from a regular MIDI keyboard. (For instance, to only have semitones on the white keys, and not to use the black keys - this would reduce the range of course.) It would make a nonsense of the repeating pattern of black and white keys though…! These would no longer have the same pattern for each octave. I'd love to try playing a keyboard set up this way.
I once played a "left-handed" piano, where all the low keys were at the right, and high keys were at the left - I reckon your suggestion would be even more confusing - but a wonderfully weird idea!
EDIT: I must stress, there is NO musical advantage to doing this - it would just be interesting… Similar suggestions have been made about having music notated on staves with equally spaced semitones, but this just doesn't make musical sense.
ANOTHER EDIT: the type of keyboard you propose would indeed make learning scales (and so how to play in different keys) similar to how a guitarist sometimes thinks about learning them. But it isn't necessarily any easier. This difference (between learning scales on the guitar and piano, and the relative difficulties of each) often comes up in my guitar lessons, so let me elaborate...
On your keyboard, you would still need to learn the patterns for (say) Major and Minor scales (two of each in fact, one starting on "white", one on "black"). Yes, you could then play all scales with these two patterns, just as a guitarist only needs a couple of patterns (or even one of each) to be able to play each scale type in all keys. Easy huh? Well, no. Because what a guitarist gains in fewer patterns to learn, they lose in reference points - apart from the fret-dots (which some guitars don't have anyway) there is nothing distinct about the positioning or patterns for scales of each different key. This would be even more of an issue on a keyboard of many octaves, which would now have no reference points, instead only a uniform alternation of black and white keys.
To sum up: yes, it is a pain learning all those different patterns for scales on the piano - I know, I'm a good guitarist but a not-so-good pianist. But, it is the fact that they all have individual patterns, which helps to make scales on the piano easily identifiable, and so learnable…
Finally, it is worth pointing out, the kind of keyboard you propose would, ironically, make it harder to play tonal music that uses major and minor scales, as the pattern of tones and semitones on a conventional keyboard reflect the pattern of tones and semitones in a major scale. A keyboard laid out with uniformly alternating black and white keys, would be best suited to playing intensely chromatic music, or music solely in whole-tones (Debussy meets Philip Glass, springs to mind...)
I'd still love to play on the kind of piano you propose, though - but don't reckon it would catch on!