Can I say "The price is going up and up as the people buy more and more."?
Yes, that's entirely idiomatic.
going down and down the road
This sounds a bit less idiomatic, perhaps.
You could certainly use this formation with going up / going down. For example you could say something like "We visited a mine. It was spooky standing there for 10 minutes with the elevator going down and down..." Or "the ski-lift kept going up and up..." - both of those sound entirely idiomatic. But those examples are when you are literally going up or down.
When we say "going down the road", the incline is often shallow or even non-existent. The "down" does not really mean "down". So to emphasise the word down via duplication in this manner sounds strange. In such a context native speaker would be unlikely to use this formation, and instead duplicate a different word. They might say something like "going on and on down the road" or "kept going and going down the road".