How dangerous is it to reveal your date of birth, and why?

The issue is not the birthday itself, the issue is a that unfortunately a lot of companies and websites are still using it for verification purposes. This is certainly bad practice and lots of companies are changing their policies just for that reason.

Banks sometimes used it for retrieving a password, but in recent years they too have changed their procedures significantly (depending on bank you use).

So to answer your question is it safe to reveal your birthdate. Preferably you only reveal it whenever it is really necessary (e.g. inquire each time if they really need it), the information is not considered secret and you will be required to disclose it on occasions for legitimate purpose. As with any personal information the best thing to do is to disclose in as few occasions as possible. On the other hand if identity theft is done and the culprit turns out to be someone being able to retrieve information or performing an action by just giving your birthday (which is easy to find). Then most likely the company will be liable for not adequately protecting your personal information or being to negligent in their verification process. Of course this will mean you will have to deal with it (which is a nuisance and very time-consuming).

The problem with revealing your birthday isn't the birthday itself, it is that you are giving people one more data point.

Reveal your birthday on site A, your relatives on site B (which gives for example mother's maiden name), your address on site C...before you know it people are able to pull together a huge amount of compiled information.

That information can then be used to hack things, either directly using password reset forms, guessing passwords, etc, or indirectly through spear phishing attacks.

For example a birthday message from an old school friend that arrives on your birthday and comes from their name would be much more convincing than a random email with a link saying "click this".

What kinds of things could an ID thief do with just with my birthday? Can he, for example, open a bank account? Recover a bank password? Open a credit card? Take a car loan?

The answers to these questions depend on space and time. By space I mean legislation in different countries and even the life style and welfare of the country someone is living in counts a lot. You can be surprised but there are lot of countries where even having a bank account is a luxury, in which case no one has to worry if his birthday is disclosed or not.

Also, when it comes to banks for example, they adapt themselves to the legislation of the countries where they are active. Also, within the same country, law changes by time so that when such information is useless for a nefarious person it can be interesting somehow in few years.

In all cases, there is no security related system that depends only on your birthday to fulfill any step of authentication because may be your birthday is already mine according to the birthday paradox. But of course, the less you reveal about yourself the safest you are. But then this will lead us to pick between being paranoid, negligent or just a wise person.

EDIT:

As you are living in USA, you know better than me that your SSN is too important. In that case, after a short research, I found that there are already some algorithms a bad person could run to guess your SSN based on your birthday and place of birth leading also to identity theft.