There are two pivotal technologies at play. One is the deflector shield, which as you will find well documented in this answer umm... deflects... any potential hazards away. Of course, what isn't very well documented is what the limits are. For example, you would expect this to be used as a military device if it were stronger. So how big of an object can you move with this?
But the deflector shield is really only useful at sub-light speeds. Full impulse is only 1/4 the speed of light. The original Enterprise was able to sustain about 1/2 the speed of light with its impulse engines but nothing approaching the speed of light for obvious reasons.
This is important because the deflector dish must emit its influnce at a speed less than the speed of light. If you are travelling faster than the speed of light, you might expect to be going faster than your deflector dish. This would be bad as you at best merely outrun your deflector dish or at worst it deflects your ship away from itself. All of these scenarios are bad.
Luckily, when it comes to faster than light travel, you don't need to worry about that because you're not travelling through space. With a warp drive, the space around you is itself travelling through space. As you ride along this distortion wave, I can't say for sure what would happen if an object happened to get bumped by your warp field, or if your warp field even exists in dimensions that matter would be able to interact with it. But if all we care about is what happens to our ship, then we don't need to worry because nothing is happening to our ship. As long as it stays within its warp bubble, it will be fine. As you can see here, the warp field appears similar in configuration to the field created by the deflector. However, since it is doing the travelling and we, relative to it, aren't moving at all, the dangers outlined above by the deflector dish are mitigated.