Can a bird fly without flight feathers?

I think the answer largely depends on which bird and whether all or just some of the flight feathers are removed.

Generally not very well. If yo remove even a few major primary feathers (from the tips of the wings) most birds will struggle. They will probably still get airbourne and not look too impinged upon, but it will take much more effort than normal so they will be flying slower, be less able to turn and will tire. Take out most of those from both sides and they probably can't take off (hence clipping wing feathers of pet birds to stop them flying away).