For one thing it can create paradoxes, like your future you deciding they didn’t like where they ended up and stopping yourself from setting out on a journey in the first place.
This is true if and only if we live in a universe with a time continuum that is not branching (i.e. a line). Instead, if we live in a time continuum that can branch (which will result in a directed acyclic graph) the act of traveling faster than light simply creates a new branch in which you go to the past and in that branch your past self never traveled.
This relates to the different interpretation of quantum mechanics. The one that has paradoxes is the Copenhagen interpretation and the one with the branching is the many-worlds interpretation.
There is also the pilot wave interpretation, but I am unsure how to map the example you proposed to that one.
I thought branches happened when a wavefunction 'collapsed' -- i.e. for every possible value of the wavefunction, you get a branch with a unique answer for 'value of the wavefunction'?
The wavefunction decoheres; it doesn't collapse. Collapse is a function of collapse theories like Copenhagen, but has never been observed in reality.
That said, decoherence doesn't branch time in the sort of absolute manner you might be thinking of. "How many timelines exist" is a question similar to "how long is a coastline"; it depends on how closely you're looking. At the smallest scales, 'timelines' interfere and mingle in a way that distinct universes really shouldn't. It's only once decoherence has gone well out of control that they can be said to be truly separate, and even then the interference never goes to zero, only approximately zero.
If we exist in a block universe, then all time-travel effects are "already" baked in. Just because something's path through the block isn't always "forward" doesn't mean it can change its path, or anything else's.
In a growing block universe, I think time travel would be impossible, but I also think that the growing block universe theory is cope.
