Tjyylli

The first thing to realise is that in most cases, whether or not to use the present perfect is a free choice: it depends on how you are choosing to relate the events to the present circumstances.

If you choose to use the perfect, you are expressing that the event which happened had some relevance to the present time. What that relevance is depends on many things: it might be that the event was very recent; it might be that it created a state which is still continuing; it might be that it is seen as part of a series of events which are still continuing; it might be that it has consequences now.

In this case, if you choose the present perfect, you are saying that being fired is relevant to the present: probably that you are in the state of having been fired, as Davo says. In this case "my boss was angry with me and I've been fired" probably means that this is very recent - today or maybe yesterday. If it was longer ago, I would have expected "and I was fired". But not necessarily: if you are choosing to emphasise the fact that you are still feeling the consequences of the firing, you might choose "I have been fired" even if it was much longer ago.

Both are fine.

...and I was fired.

This explains what happened in the past - you were fired.

...and I have been fired.

This explains your current situation - you are in a state of having been fired.

Can I also say "My boss was furious with me and I have been fired."?

That's an error, but it's not a grammatical error, more of a style error. "My boss was furious" is simple past. "I have been fired" is present perfect. Present perfect is a mixture of past and present; it discusses things that happened in the past, but does so with respect to the current situation. "I have been fired" means "My current state is fired". While the firing happened in the past, the focus is on the current state of unemployment. Thus, this breaks up the connection between the two clauses. "My boss was furious with me and I was fired" presents the two clauses as two connected facts: my boss was furious -> my firing resulted. "My boss was furious with me and I have been fired" breaks up the flow and makes these sound like two random facts you've decided to put in one sentence, rather than causally related.

Could I say "What to do right after you go fired?" too?

No, "go" can't be used that way. You can say "What to do right after you get fired?" or "What to do right after you are fired?"

You might want to post this on the German SE to get the perspective of people fluent in both languages as to how they compare.

Your sentence is set in the past "By the time I got to the office...". That is why the past tense is used, I think. There is also a sequence of events:
1.the meeting began 2.you got there late 3.boss was furious 4.you got fired. Past simple is also used for sequence in the past.

"I have been fired" could be used to announce this recent fact (announcement of news) which is on your mind now because it affects the present = you have to look for a new job.

"when you have been fired" is a passive sentence. Someone else has fired you, you have been fired by your boss.

The answer depends on what language you are asking about.

In English, "...have been..." is a statement about the present as well as the past. It talks not only about what happened, but also about what the state of affairs is now. For instance, in English, "The software has been installed" means "The software is in a state of having-been-installed" - or, to be less eccentric about it, it means that both (1) "The software was installed" and (2) "The software is still installed". On the other hand, in English, "...was..." is a statement purely about the past. It says nothing about the present.

In American, this distinction is rarely made. The form "...have been..." is rarely used, and "...was..." usually replaces it. This often causes confusion. For instance, when an American-speaking computer pops up a message saying "The software was installed", an English-speaker will think "Why did it not say the software has been installed? Does it mean that the software was installed but then something went wrong afterwards?".

So in your case, as a foreigner, the best thing is to learn the more precise distinction - thus, using "have been" if you have been fired and are still fired and it happened recently, and "was" if you are talking about a more distant past, or you already have another job. This is correct English, and speakers of American will understand it without thinking it strange.

(Interestingly, there is exactly the same Atlantic split in Spanish: in Europe, me han despedido means it was recent and I haven't got another job since, while me dispidieron means it was further in the past or I do have another job; in Latin America, they use me dispidieron for everything).

I agree with (most) other posters that both sentences are grammatically fine. However, I think there is an additional distinction which hasn't yet been mentioned. In

my boss was furious with me and I was fired

you are using exactly the same tense for both things, and this suggests that they happened at the same time.

However, in

my boss was furious with me and I have been fired

you are using the past progressive for the fury (this was happening then, when you were late for the meeting) followed by the present perfect for the firing (that has happened by now). The suggestion is that your boss was furious, and at some point between then and now you got fired.