If you're an average-healthy person from (near) sea level, how high may you climb a mountain before needing to stay at the altitude to acclimatise? Is it alright to go/drive up a 9000 ft (2750 m) mountain in one day or would it be dangerous?
People react differently to altitude, and it is not always those you would predict who have trouble at altitude. Thus, my advice is to take it slower than zero to 9,000 feet in one day unless you have done it before with no problems.
Driving up to 9,000 feet is one thing, because you can quickly retreat. Hiking up, especially if you spend the night, is more problematic. Moreover, hiking up 9,000 feet in one day, especially with a pack, unless you have done it before, is probably not the best plan, independent of the altitude.
Consider that the guided hikes up Rainier, for example, which have a 9,000 foot elevation change (5,400 feet to 14,400 feet), do it in two days. Do you want to drive up to Paradise Inn (5,400 feet) from sea level and immediately hike up to Camp Muir (10,000 feet) (which is a non-technical, easy hike)? My advice, based on personal experience, and the level of familiarity with altitude that your question suggests is "no".
As a rule of thumb, high altitude sickness can start at around 2500m. However, for most people this will not be serious yet (maybe except for getting out of breath quickly), especially when only there for a short time. Otherwise, skiing, hiking and driving mountain passes would be a lot more problematic.
For most people, problems start when they stay overnight at around 3000m. Keep in mind that things start to change quickly when you go further up. The step between 2500m and 3000m might be the same altitude difference as going from 3000m to 3500m, but the effect is vastly different.
Given that the symptoms are mild at medium altitudes and the body will adapt quickly, the best way is to try things out and see how you feel. If you go higher than 3500m - especially overnight - I would recommend some acclimatization beforehand. Above those altitudes, it is recommended to limit your sleeping altitude to an increase of around 500 vertical meters per day and plan for sufficient rest days
The highest places for mass tourism in Switzerland are Jungfraujoch (3466m / 11370ft) and Titlis (3238m / 10620ft). People from all over the world - in particular from far-eastern countries - go there in just one day, with their hotels at maybe 400m. If people would regularly get health problems or pass out during that trip, those trips wouldn't be so famous and crowded.
All the tourists descend at the same day though (there's no public accomodation there) and they're not supposed to hike or do other intense activities there, so this might also reduce the impact. And the railway up to Jungfraujoch makes two short stops during ascent - officially to offer a great view from a station in a cave, but in fact to acclimatise the passengers.
In Colorado, you can drive to almost the top of Mt Evans. There is a parking lot about 100 feet feet below the 14,271 ft (4,350m) peak. I have been there repeatedly and have taken a number of my relatives. There are taller mountains than this in Colorado, but this is the highest road.
One of the most noticeable things, and the reason I have taken my relatives up, is that I got noticeably dizzy as soon as I got out of my car and stood up. That's actually one of the tourist attractions -- noticeable altitude sickness. It was enough to feel, but not enough to stop me from actually doing anything. I'm not in the best shape, and the remaining 1/4 mile (400m) trail to climb the remaining ~100ft of elevation to the top is pretty steep, but I was still able to make it up to the top.
Some of the relatives that I took up there flew from near sea-level to Denver (5280ft/1600m) and went with me the rest of the way up the next day.
Based on this experience, I think that at least 15000 feet is not an issue. This covers the entire 48 contiguous United States, and something like 99% of the entire Earth.
You're asking a question that doesn't have a real answer because it depends on the person.
I have driven 5,000' to 12,000' without any issue. I have drive + hike from 3,500' to 12,000' without problems. I have drive + hike from 3,500' to 10,000'+ many times, no problems. I have hiked 5,000' to 18,000' over thee and a half days, no problems--even though nearly 80% of our group had to turn back at 15,500'.
I would hesitate to drive my wife from 5,000' to 12,000', though, and would keep an eye out for symptoms and head down immediately if it were an issue.
I would not even consider driving her sister from 5,000' to 12,000'.
I often go from 300 m height to 2500 - 2800 m in a day, sometimes I sleep at 2500 - 2700m without any problems. I think there is enough oxygen, pressure high enough and if you are average health you shouldn't be concerned.
In my country people often go from low elevation to 2800m in a day and I don't see anyone making complains or talking about acclimatization.
If you would go above 3000 - 3900 m then yes you should acclimatize for a day.
It's almost less about being at a certain altitude than exerting yourself at a particular altitude. And it depends on the person. If you're just driving up to 9000ft, can't imagine it making a huge difference, unless you're sensitive.
Climbing a mountain is a different matter. I lived in Vail for years (8250 feet, with area peaks going over 13,000ft). I remember going for a run my first week there, and realizing there would be an adjustment period. That's when I noticed. It wasn't just being there, or taking slow and steady walks, it was exerting myself. I was out of breath to the point it hurt to breathe in less than a mile (about a half km). Even after living there for years and fully acclimating, while regularly being active, the altitude has an effect. You notice, when you're tired and panting, there is less oxygen getting to your body. Recovery can also take longer, at least if you're not used to it. My first climb in Vail, I had to take the whole next day to rest as my legs were just shot.
So, if you're climbing a mountain... depends on you. I hear of people going from sea level to Colorado just for a weekend to run an ultramarathon, and some of those people report no ill effects (though they are clearly in excellent physical shape to do this, and have been training for the extreme lengths of running at sea level, still). Someone else doing that could pass out after a few miles. In Denver, at about 5200ft, I never really noticed much. So, I'd say somewhere around the 7000ft mark is where you may start to feel it more, on a guess.