After a verb with connotation (so not as generic as tell/say) it's best to keep "that". I regret to tell you that. But as mentioned by others, even better would be: I regret to inform you that...

More common is : "I regret to tell you, you're fired"

It would make sense but doesn't sound the best. You would want to keep it past tense and combine the second 'you' with 'have' to make "you've" that would make: "I regret to inform you that you've been fired" but the most gramatically correct way of saying this would be: "I regret to inform you that you have been relieved of your position". "I regret to inform you" is quite formal and "You are fired" is quite aggressive/informal so the above would be the most natural sounding as it's formal all the way through. Hope this was helpful. *Edited to correct minor errors

To allow the sentence to flow the addition of 'that' between the yous will help. 'I regret to tell you that you are fired' or 'I regret to tell you, you are fired'