Would it be illegal to use a symbol as your logo? For instance, could the ohm symbol be the logo for a yoga studio? Could the peace sign be a logo for a non profit organization? Could a unique cultural symbol such as the Native American Zia Sun symbol be the logo for a shirt company?
This question might seem innocent at first, but has a lot more to it than one might think. A first reaction would be to say 'of course you can use them, they're not owned by anyone'. But it really isn't that simple. Most things have been designed by someone at some point in time, so even though they are widely used it doesn't necessarily mean you can use them for a (commercial) logo. Let's just take a look at your thee examples + one bonus:
Omega or Ω (the letter used to indicate electrical resistance - Ohm) is not a symbol but just the Greek capital letter O. The use of letters is not forbidden in logos (luckily), no matter the language, so there should be no problem. There are a lot of companies out there who use Greek letters in a logo, not in the least Greek companies. Of course, a lettertype (font) may be legally protected, so a specific shape of letter might get you in legal trouble, but not the letter itself.
I'm going to assume with 'peace sign' you mean ☮ . This is actually an interesting case, since it was designed in the fifties for the Nuclear Disarmament movement by Gerald Holtom. But since the designer has specifically not copyrighted it, it has become widely used as a general peace sign. In such, it's probably safe to use in a (commercial) logo. 1
The Zia Sun symbol is another special case. It's so old, it's automatically in the Public Domain and you are allowed to use it in a logo. That said, it still holds a significant cultural meaning to the Pueblo people. They have specifically requested you ask permission from them to use the symbol. They cannot actually take legal action against you using it, but they can make your life very hard. 2, 3
The Swastika, much like the Zia Sun symbol, is an ancient symbol used by many cultures and in many different contexts, and as such is in the Public Domain. Originally, it meant something very positive, and in a lot of cultures still does. But of course, in Western society it has taken on a meaning of extreme hatred and racism. 4
Each and every symbol and shape is unique, has a history, invokes associations, has lovers and haters, can have multiple meanings, etc. By using an existing symbol, you can create a kind of shorthand to invoke the same feelings and associations for your brand. But meanings and associations can change. People may grow tired of symbol overuse. There may be legal limitations that are not immediately obvious. It may be legal for you to use a symbol (please seek legal counsel for a definitive answer), but people or time could still make you regret it.
In conclusion: Is it a good idea to use a symbol in a logo? While there may be instances this may be true, for the most part I would say: no, it's not. You're probably better of designing an original logo, using shapes, colour and words to invoke the same feelings in your target audience.
Having just done research on trademark law for my own company's logo, I think the question about whether or not it is illegal to use a symbol as a logo is definitely something that needs to be answered by a lawyer.
However, if the question is more about whether or not it is a good idea to use a symbol like the Om symbol or a Zia Sun Symbol as a logo--well, the point of a logo is really to help potential customers/consumers connect the products or services that the company provides with that company. If there is a chance that the symbol will be seen by customers in relation to other situations that are not related to the company, there is a strong potential that the brand for that company becomes diluted or even damaged.
This is not as likely to occur if you incorporate that symbol into a personalized logo that includes elements of the company name, etc., but should still be a consideration.
A concrete (hypothetical) example: Let's say I start a company that sells herbal supplies that are based on herbal lore I learned from my native grandmother, who was Zuni. I might want to use the Zuni sun symbol in my logo as a way to illustrate this connection with traditional Zuni knowledge.
Another person starts a business making pottery in the Zuni style. They also use the Zuni Sun Symbol as their logo. But, they aren't very good potters, and they begin to earn a reputation for making pottery that breaks easily.
Both businesses decide to sponsor an event, and they are asked to provide their logo for the event catalog. My herbal company shares a visual identity with the shoddy pottery company, and the negative reputation can easily (although unfairly) roll over and impact the perception of my business.
TL;DR Using a common symbol, particularly on its own rather than as an element within a larger design, increases the potential for having your brand identify damaged or diluted.
And all of this is also subject to consideration of the issues raised in @piebie's response.