All three of these are acceptable, and indicate that the project will be done before or during March, but have slightly different connotations
1) "The project will have already finished by March" suggests that this is being discussed in the context of something that happens later than March. The "will have" construction implies looking backward from a future point in time, and the "already" indicates that March is sometime before that future point. This construction, overall, suggests that the project is going to finish too early for something else that will happen later. This would be an appropriate phrasing if someone was suggesting a project team meeting in April, for example.
2) "The project will have finished by March" is again looking backward from a future time. Very similar to the first phrasing, but without the "already" there's less emphasis on the discrepancy between the later time and when the project finishes.
3) "The project will finish by March" is the simplest construction, simply stating by what time the project will finish. There is no relationship to some future time beyond March, this is just a forward-looking statement.