Based just on TV, there's no definite answer either way. There are clues that could argue in each direction. For example, on the one hand, the Monk and the Master seem quite different in motivation and approach. On the other, the Doctor and the War Chief immediately recognise each other, and the War Chief does dress very similar to the Master, especially when you compare him to other Time Lords we meet later. But there's absolutely no solid evidence either way.
The novels, audios, and comics all make it quite clear that they're different. Most importantly, in the BBC novel 'Divided Loyalties', they all went to school with each other, and were named Koschei (the Master), Mortimus (the Monk) and Magnus (the War Chief).
Furthermore, when the Master (as Missy) finally met the Monk in "Divorced, Beheaded, Regenerated" (Missy audio story), they both acted as if they were separate people rather than different incarnations of the same Time Lord.
On the other hand, the FASA role-playing game and the novel Time's Champion make it just as clear that the Monk is definitely an earlier incarnation of the Master, and the War Chief probably is another.
Since there is no authority on what is and isn't canon, that's about all that can be said.
For arguments for and against the identification, based on in-universe, out-of-universe, and canonicity considerations, see the Talk page.
Writers have occasionally played with this confusion in cute references. For example, in a short story, Death commented that she sometimes had trouble telling her two most recent champions apart—a joke for fans who knew about the RPG, and that the Monk and the Master had both been her champions (in the Virgin novels and the Big Finish audios, respectively).