Some story content will be locked out if you have not already completed it.
Beginning with Knights of the Fallen Empire, starting a new storyline will automatically complete previous story content, locking you out from playing it. Story decisions in the skipped content will be made automatically based on your faction and alignment. Non-story content - outdoor zones, flashpoints, PvP, operations, etc. - is unaffected. You can continue to visit the same zones and participate in the same group content as before.
What story is skipped depends on what you're starting:
Knights of the Fallen Empire: your class's original 1-50 class story, Makeb, the Forged Alliances storyline, Rishi, Yavin IV, and Ziost.
Knights of the Eternal Throne: all of the above and Knights of the Fallen Empire chapters 1-16.
Iokath: all of the above and Knights of the Eternal Throne chapters 1-9.
Jedi Under Siege: all of the above, the introduction to Iokath, and the Fractured Alliance storyline.
This list is from memory, so it may have a few mistakes. When you begin a new storyline, the game will prompt you with the stories that will be autocompleted. You should read that prompt carefully and make sure you've completed any of the listed stories that you want to play through.
The main chapters of Knights of the Fallen Empire and Eternal Throne can be replayed any time you like through the Galactic Command window. You can select choices other than the default while playing a chapter, but future chapters will not "remember" them.
A note on companions: when you start Knights of the Fallen Empire, your original companions will leave you and you'll acquire new companions over the course of the story. Once you have access to Odessen (by completing or skipping chapter 9 of Knights of the Fallen Empire) you can use the Companion Locator Terminal in the mission hub to reacquire any missing companions from your class story if you wish to. It's worth noting that all companions can now perform all combat roles, so don't worry about being stuck without a tank or healer.
Finally, this is a purely per-character decision. Each of your characters can progress into Knights of the Fallen Empire at their own pace without affecting your other characters at all.