So, I did some digging on this, and I found out how Red Bull is actually structured as a company.
The company that controls Red Bull (the brand), the Austrian registered Red Bull GmbH, is owned 51% by the Thai Chalerm Yoovidhya and 49% by Mark Mateschitz, the son of Dietrich Mateschitz. So, while at first glance, the Thai side of the operation always had control of the company, in fact, Dietrich Mateschitz had an agreement that gave him 51% of the voting rights in the company.
So, in the UK, Red Bull Racing Ltd (i.e. the company that came from the original Stewart F1 team and operates the F1 team known as Red Bull) has two directors: Horner and Helmut Marko, with Horner as the "director" (i.e. de facto CEO) and Marko as a board member. That company in turn is owned by Red Bull Technology Ltd, which is the parent company for all of Red Bull's motorsport interests in F1. Red Bull Technology has two directors, Horner as the de facto CEO and the finance director Alastair Rew.
Now, Red Bull Technology is 100% owned by Red Bull GmbH, so when Dietrich Mateschitz was alive, he was de facto in control of Red Bull Racing. But in fact, there was no agreement made between Dietrich Mateschitz and the Thai side of the operation for this control to pass to Mark Mateschitz, meaning that the Thai side of the operation is now in control of the motorsport operations.
So, to answer your question: yes, they can fire him. However, Horner has excellent relations with the Thai side of the operation, meaning that even if the Austrians hate him, there's nearly nothing they can do about it. There's a lot of political moves going on within Red Bull GmbH, but as long as he retains the support of Chalerm Yoovidhya, he's untouchable. It's very likely that he sees Horner as a balance to the Austrian corporate types that run the company, especially as the F1 team is a massively important part of the Red Bull brand and business.
The Thais in particular have been said to be taking much more of an active role in the business, meaning that this is really about a power struggle at the top. It's well known that the Austrians want to get control over Red Bull Technology, as the company has been operating very much at arms length from Red Bull GmbH. Dietrich Mateschitz didn't mind or even care, because Horner was delivering, but now with him gone, the knives are out.
But there is something else to all of this. Horner's patron was Bernie Ecclestone, and it would be very hard to remove him without him doing something wrong. Liberty Global won't want him to go anywhere as he's so media friendly, and most F1 team owners would probably agree that Horner is excellent for the sport. This is exactly why Toto and others are getting so frustrated with Red Bull GmbH's handling of the whole situation, because they simply want to compete without this ridiculous situation hanging over them. You'll notice that they haven't said anything bad about Horner, it's all criticism aimed at the parent company.
