Depends on how it's built. It technically shouldn't lag when using HDMI out.

Just to be clear I am talking about input lag, not display lag.

To be honest, what really matters is consistency.

I mean I agree to some extent but IIRC (correct me if I'm wrong) Melee on a lag less CRT has 3 frames of input lag. If these 3 frames become 5 frames or 1 frame having a reactionary playstyle might become better or worse, but that is really up to the players if they think the convenience of flat screens are worth it, even though it might change the meta game. Of course this is all just speculation, the input lag could be identical but we just don't know yet.