Would a director revive a dead character for a sequel?

Probably the most famous example I can think of for this is Terminator 2: Judgement Day.

Despite his rather untimely demise in the previous film, Arnie is back albeit technically as a different terminator in the second film - however I fully admit I don't know enough about the film to know if this was always intended, or simply happened due to the financial implications of having Arnold.

Of course, an issue with your question is what you mean by killing a character. For example, pretend Lord of the Rings was a movie-only experience and no books had been written. I think most people would have looked at the first movie, and Gandalf's fall, and assumed death - but all that needed was a few clips to expand on the scene and everything changed.

In the movie you referenced, Scarface, given the fact it's not a sci-fi movie and there's no silly futuristic technology capable of doing anything, it would be difficult to imagine how anyone could return from that ending. But if he had been shot once and fallen off a cliff, since we never see the moment of death, he could certainly have returned - although whether or not he would have "died" in the way you meant would obviously be questionable.

Finally, although it's not quite what you asked, director's can leave a character in limbo. A famous example of that was Han Solo in the Empire Strikes Back. The story goes that Harrison Ford hadn't agreed to sign on for the final film yet. George Lucas, wanting to simply freeze time until the contractual negotiations had been completed and he knew one way or another did just that - he froze time, or rather Han Solo in the carbonite. That meant they could either kill him off if the negotiations didn't work or revive him, as they did, in Return of the Jedi.

Edit

Forgot to include the legendary Kris Kristofferson, playing Abraham Whistler, in the Blade films. Killed himself in the first one after becoming infected, pops right back up again in the second (although we never see the death in the first film, it is strongly implied as he is given a gun by Wesley Snipes who then walks away as a bullet fires).

I accept that it's not a movie but perhaps the canonical example of this is the return of Sherlock Holmes.

Wanting to devote more time to his serious novels, Arthur Conan Doyle killed off Holmes by having him plunge to his death from the Reichenbach Falls locked in a struggle to the death with his arch-enemy Moriarty.

Eight years later in real life (but three in the canon) after Doyle bowed to increasing public pressure, Holmes reappeared and explained to a shocked Watson that he had only faked his death in to fool his enemies.

"The Adventure of the Empty House" marks the beginning of the second set of stories, which Doyle continued to write until 1927, three years before his death.

Bobby Ewing in Dallas is a fine example

Most often when they want to revive a character they start the sequel with an explanation why the character didn't die in the end. It doesn't have to make much sense of course, but well. In Death Race (2 and 3) for example it is shown that the main character didn't die and another guy was wearing the mask of the main character during the fatal race.

More often the dead/end of the main character is more mythical or unclear to keep the option of sequel open. Like in Crank.

In Crank the main character falls from a helicopter. He hits the ground/car, supposedly dead. But two heartbeats can be heard as the film fades to black and he blinks, the movie. Hence, the let the option open. Then, in Crank 2 (High voltage) he is alive again in and in "normal healthy" condition.

It is of course possible to revive a dead character in the sequel but audience are not that stupid to believe such revivals.. Wat I have seen in many movies is that the same actor comes in the sequel as the dead character's son and the story is altered accordingly.

In the movie Scarface, what they could have done for a sequel is that Tony Montana's wife was pregnant at the time of his death and Al Pacino comes in as his son in the sequel.

Tony Montana himself was brought back for a video-game sequel. I know you were asking about movies, but since you used Scarface as an example, I think this might be of interest.

Scarface: The World is Yours is a quasi-sequel beginning with the film's final scene.

The game begins in the film's final scene, with the mansion of Tony Montana (voiced by André Sogliuzzo) being raided by assassins sent by Alejandro Sosa (Robert Davi). In a point of divergence from the film, Tony kills Sosa's assassin the Skull, that was to slay him. With the assistance of some of his surviving employees, Tony manages to escape just as the DEA and the Miami Police Department arrive to gun down the remaining hitmen.

Of further interest re: the filmmakers:

The developers originally asked Oliver Stone, the film's screenwriter, to write the script for the game. When Stone declined, they approached American screenwriter David McKenna, known for writing American History X and Blow, films which also feature criminal antiheroes. McKenna accepted after seeing early gameplay visualisations. As a fan of the film, he wanted to emphasise the over-the-top humour he perceived in the character of Tony Montana.1

Although Al Pacino lent his image to the game, he did not provide the voice for the Tony Montana character, as his voice had deepened considerably since the production of the film, due to his years of heavy smoking. Instead, actor André Sogliuzzo (who was selected by Pacino) imitated Al Pacino's voice.

Yes, pretty much anyone can be brought back for sequels. Sometimes it may be in another medium, though.

My favorite example of this is Chow Yun Fat in John Woo's A Better Tomorrow II (screenplay by Tsui Hark). Fat dies at the end of the first A Better Tomorrow, but he comes back in A Better Tomorrow II as the twin brother of the protagonist in the first film. Sure, they're technically different characters, but their demeanor/action-style is pretty much the same. And the real kicker is that, as silly as this plot device sounds, A Better Tomorrow II is a better film that its predecessor in near every way.

Aside from that film, and the films mentioned above, sure. It happens all the time. Especially in horror films. Take Freddy Kreuger in A Nightmare on Elm Street, for example. Sure, he's the antagonist in the first film, but as he became the box office draw, he really became the series' protagonist.

From previous comments:

(An) example would be Crank, or to be exact Crank 2, making him alive again after dropping out of a helicopter up high. That is to some degree set in a present day non-SciFi-Fantasy reality, yet a quite weird and stretched reality, though. – Napoleon Wilson

AND

Another example would be the Alien series, where Ripley is reborn either from hyper-sleep or DNA resurrection throughout the series. - Paulster2

And a few other examples:

"Curly" from City Slickers being brought back (using the same actor) as a twin brother. Usually a lame gambit used for soap operas.

Agent Smith was killed in The Matrix. Brought back for parts 2 and 3.

Spock gave his life in Start Trek II: The Wrath of Khan "for the many" only to be brought back, ruining the meaning of his selfless sacrifice.

Elektra. Ugh, I almost don't even want to get into that awful nightmare.

Ramirez, played by Sean Connery, dies in Highlander, and returns to life, with no explanation whatever, in Highlander II. Both films were directed by Russell Mulcahy.