On Elder Scrolls Oblivion and Skyrim you can join all factions and become the leader. I'm also playing Morrowind and I'd like to know if I can do the same, because I've read that some guild missions conflict with other guild missions.
Yes certain factions conflict with each other. I tried to keep the following list relatively spoiler-free so I apologize for vagueness.
The Great Houses of Morrowind are mutually exclusive and as such, you can only join one of the three choices (Hlaalu, Telvanni, or Redoran) without the use of a bug.
The Fighters Guild quest "The Code Book" is in direct conflict with the Thieves Guild and will make it so you can not join the Thieves Guild. There is a way to get around this as explained on the wiki page linked below.
Any Fighters Guild quest offered by Sjoring Hard-Heart make the Thieves Guild hate you at a minimum or kicked out of it at a maximum.
A Tribunal Temple quest, "Slay Raxle Berne", will block off being able to join the Berne vampire clan.
The last one I can find deals with the Mages Guild. One of the final quests they give you will make it so you can no longer be a part of House Telvanni.
There is an entire wiki page dedicated to quest timing that talks about conflicting quests. You can find it here.