There's room to do (almost) whatever you want
Drugs (psychoactive or no) vary extremely widely in terms of:
- whether or not they ever induce tolerance and how fast they do if so
- whether they have an obvious withdrawl syndrome
- how debilitating or dangerous the withdrawal is, if any
- whether or not it's realistically possible to overdose and how easy it is if so
The drug ziconotide is sometimes used in hospitals in place of morphine because it's a strong painkiller that appears not to induce any obvious tolerance even after months of constant use. In line with this, someone can stop using it abruptly "without any concern for the development of withdrawal symptoms." (The main problem with it is that it has to be given via spinal tap.)
Drugs that actually induce dependence, not just tolerance, often take a while to do so, like a month of daily use. There are some drugs out there that induce tolerance within a few days (tachypylaxis), although this doesn't necessarily come along with a withdrawal syndrome if the person stops taking the drug at that point.
If a drug does cause withdrawal, the symptoms vary widely between drugs, both in terms of their severety and how long the withdrawal lasts. I've gotten the impression that drug withdrawal tends to take sometime between a few days and a few weeks for most drugs. Some come with relatively mild symptoms, like caffeine, which causes grogginess and headaches; others come with more severe symptoms, like SSRIs, which can cause intense nausea and vomiting during the withdrawal. Even worse, a drug I take every day for migraine prophylaxis, propranalol, has a black box warning telling you not to stop taking it abruptly because it can cause dangerous heart palpitations and even heart attack. Alcohol withdrawal is also infamous for its danger—it can involve seizures that can lead to brain damage and death.
As for overdose, some relatively popular recreational drugs are non-toxic enough that you would struggle to chemically injure yourself even if you took doses thousands of times larger than is recreationally common. Other popular recreational drugs are remarkably toxic, like nicotine—about 0.5–1 g of nicotine, taken orally, is enough to kill most adults. Smoking a cigarette only gives 2 mg or so, though, so nicotine overdose isn't very common. A more serious case of active vs. toxic dose comes up with acetaminophen, i.e. Tylenol, where a common over-the-counter dose of 0.5–1 g is worrisomely close to the maximum recommended 24-hour dose for an average adult, 4 g. Hundreds of people die from accidental Tylenol overdose every year in the U.S. alone. Perhaps we're fortunate that it doesn't come with a pleasant high.
All of this is to say that I think you have room to have your mind-control sedative have whatever properties you please as far as tolerance goes. There's definitely room to have it never induce tolerance. You could also have it induce tolerance very slowly, ramping up gradually over months or years, or have it induce tolerance very quickly, over the course of a few days. You could give it a viciously unpleasant or even deadly withdrawal syndrome, a mildly uncomfortable withdrawal syndrome, or no withdrawal syndrome. You could have it be highly toxic, so that it's scarily easy to overdose on, or have the toxic dose be so far away from the psychoactive dose that overdose is unknown in the medical literature, or anything in-between. The human brain is famously complicated.
One thing I will say is that the medical reality rather favors the government in your setting, I think. There are lots of drugs that keep working to some extent as you keep taking them, but you do develop a dependence, and they can be deadly to stop taking suddenly (again propranalol is such a drug). If the sedative was like this, people might be afraid not to take it once they had gotten started on it. You can wean yourself off of drugs like that by gradually reducing the dose, but if the only way to get the drug was to go to a government clinic and have them administer it or something, you wouldn't necessarily have that much control over the dose. That could keep people in thrall to the sedative just out of grim necessity.
Alternatively, you could have the drug not induce tolerance, be quite non-toxic, and very pleasant. I don't think you would need to try very hard to coerce people to take a drug like that—just make it widely, freely available and present it as a healthy habit. Soma in Aldous Huxley's Brave New World is perhaps the "textbook" example of this in English-language fiction.
To be honest, I think that might be only realistic way to have a drug keep people from revolting without also intensely disabling them. If it makes them happy, they'll put up with a lot. If they're just sedated, it wouldn't necessarily make them not want to revolt, so you would have to sedate them enough that they were too sleepy to drive or work or anything too. I think it would be hard to keep the society going that way. So, a drug that makes everyone happy and satisfied seem more plausible to me as a way of preventing revolt. We don't really know of any drugs that are true "mind control agents," in terms of making people pliant and suggestible but otherwise capable of acting normally (the CIA and KGB looked very hard for things like that during the Cold War…). I suppose we don't know of anything quite like soma, either, but somehow it seems more plausible to me, if nothing else because we do know of many drugs that induce various flavors of euphoria.
I think the only thing that's not likely is to have it so that people develop complete tolerance that never goes away after taking it for a while. Generally, if someone stops taking a drug that induces tolerance, their tolerance dissipates gradually. If the drug was dangerously toxic, though, I suppose it's possible that enough people would die from taking it to exert selective pressure on the population over time. If some people are genetically immune to the drug, they won't be at risk of overdose (or at least not in the same way), so they would be more likely to live and pass their genes on. For this to result in widespread immunity, either the gene would have to be very easy to inherit once the mutation arose somewhere or a lot of non-immune people would need to die off in a short timespan (like if the government introduced the sedative widely in a reckless fashion).
"Pratical" complete tolerance is more feasible, though. If the drug has a toxic dose near the active dose and induces tolerance over time, it could get to the point that the active and toxic dose were practically overlapping as someone kept taking it. At this point, a non-toxic dose wouldn't have much of an effect on the person in psychological terms, and a toxic dose might kill them, so the only thing they would get out of taking it is staving off withdrawal. In the "forcing everyone to take it to prevent deadly withdrawal" scenario, this kind of effect could throw a wrench in the government's plans once enough people got to that point, especially if nobody really noticed what was happening at first.