Usually they are designed asymmetrically for a reason. If properly designed, there are no disadvantages.
The Rutan Boomerang is an example and it flies quite conventionally. It is not a warplane, but the Blohm und Voss BV.141 was designed that way for good visibility forward and all around as a WWII observation aircraft.
In the case of these two, there are no disadvantages in that the planes performed as designed and exhibited no unusual flight characteristics. If there were any asymmetric planes that were NOT well designed, I am not aware of them. If such birds exist / existed, then the disadvantages might include things such as if the tail were asymmetric and not well designed, then pulling up might also induce roll. So control coupling issues might arise. I don't think anyone would place a signle engine so far off-center as to vause spin problem near the stall, but that could arise depending on the design.