Take a hypothetical example. If I only use the middle two octaves in the piano for some piece, would the piano sound badly if some of the keys in the other octaves are out of tune. Or would the piano sound better if all keys are in tune because of sympathetic vibration? Thanks
The sympathetic vibration is the thing. When a key is pressed, and the damper pedal depressed, all the dampers move away from all the strings. Not only do the octaves of that note ring out sympathetically, but also, to a lesser extent, the other notes which are octave copies of the original's harmonics. So, yes, it would sound weird as in nowhere as rich.
To my best knowledge, the harp and soundboard would become misshapen and it would cause unnecessary stress on the instrument to leave it "half tuned" like you're describing. This would make it harder to get it tuned later on as it would require extra tunings to get the harp back into shape.
But hey, if you don't care about the instrument too much, do what's practical. It might even have a unique quality to the sound.
It wouldn't sound quite the same as a well-tuned piano. But probably not a whole lot different. Assuming the untuned strings are still at SOME pitch in the right ballpark, they'll still resonate when the pedal lifts their dampers. It doesn't matter if the one labelled 'C' is actually resonating at some other pitch!