The OP doesn't specify a country, but given that Tyson is a citizen of the USA and the quote concerns a topic of significant political interest in the US, I'll look at this from the historical American perspective.
The Founding Fathers enshrined the right to speech uninhibited by the government in the 1st Amendment. They believed that true (political) freedom required a free and open exchange in the "marketplace of ideas". They also believed that over time only the most meritorious ideas would rise to lasting prominence, and that this freedom of speech was necessary to ensure that merit would be the dominant factor: ideas would neither rise nor fall (in the long run) due to other factors. This conception of free speech in the US has also been used in a number of Supreme Court decisions over the centuries. In this idealized sense, "false beliefs" would not have been considered a threat in any way whatsoever. Over time the merits of truth would drown out the false beliefs, and the people would naturally move to democratically select truths over falsehoods.
The question, in modern times, then changes to asking how close we are to this idealized scenario: do we have free and open exchange on the marketplace of ideas, impeded only by the merits of our ideas? That's extremely difficult to answer, and hits on a number of divisive political issues.
For example, the liberal side of the political spectrum tends to decry the Citizens United decision, which allowed essentially unfettered and limitless political donations from corporations. The liberal side sees this as tilting things in favor of the ultra wealthy and corporate interests: ideas are rising to the top not because of their merits, but because of the wealth behind them, while other ideas sink because their supporters are (relatively) too poor to compete.
On the other hand, the conservative side of the spectrum tends to rankle at the effects of "political correctness" and public reactions to alleged scandals that they feel violates presumptions of innocence, or otherwise relies on little more than one person's claims. Because they cannot use the words they want without being attacked and vilified for the words rather than for the ideas (from their perspective, at least), they feel disadvantaged and marginalized. They feel they cannot speak freely, and must dance around the PC landmines or be shunned.
Those are just two items, one from each "side" of the conservative/liberal divide (and I try not to pass judgment on them here), but there are many more. And there are deeper and more subtle issues that contribute to the matter, which are largely independent of political leanings and individual perceptions of their place in the marketplace of ideas.
The internet, and mass media in general, has provided a vehicle for the exchange of ideas that was likely well beyond the imagination of the Founding Fathers. It is now much easier to find and associate with people who share your ideas, no matter how unusual the may be. It then becomes much easier to constantly hear support and reassertions of these ideas, which helps to hedge out competing ideas: an echo chamber.
The sheer wealth of information, and the ease with which information is accessed, is also on a scale that was unimaginable even just a few decades ago. It is relatively easy to find some minutiae to push an agenda, and difficult to quickly find a concise counterpoint. A favorite plaything of Creationists is to try to find some tiny thing that science does not yet entirely understand, or it does but is difficult to explain to laymen—e.g. mathematicians have a phrase "almost surely", which has a very precise and unambiguous meaning; but it sure doesn't sound like it's precise—, and harp it as evidence that science is false and a failure. This wholly disregards the incomprehensibly huge volumes of evidence science has for the large swathe of things it does understand pretty thoroughly, as well as the general principle that science is always growing and changing.
In this way an idea can seep into the minds of people, who then may surround themselves with the like-minded to the exclusion of all other ideas in perpetuity. Now they are no longer engaging in a free exchange in a marketplace of ideas at all.
Now it's hard to say if that all sounds worse than it is, or how different it really is from the past (we've had insular religious groups many times, after all), and what the true long-run future looks like. The point I think to draw is: ideas themselves, regardless of what objective truth they may possess or lack, are not a threat to (the American conception of) democracy. What matter is the marketplace of ideas: how we convey, adapt, and change our ideas. "False beliefs" in a healthy and uninhibited marketplace of ideas are not seen as a threat; they are a natural and welcome participant. But an unhealthy or inhibited marketplace is a significant threat, regardless of what ideas are peddled in it.