Ikaruga

I can confirm that neither player is allowed to fire a shot. Furthermore you are not allowed to die even once.
Follow the above conditions and you should get the "dot eater" rank on the grade screen after the boss, along with the achievement and a little message saying that there are "dot eater" leaderboards available.

I died once and my partner and I both got this in local co-op, so that's incorrect.
But yes, if you play this co-op, neither player is allowed to fire a shot, or it negates the achievement for both of you.

i can confirm that you are allowed to die as i died once against the boss and the achievement still unlocked

@Liquid - although the guide misses this important tidbit you can actually do this on "Easy" difficulty with continues, extra lives or whatever else you want enabled. Makes this a walk in the park!

Ok so I had to do this twice, make sure you actually have a button mapped out for the fire. I turned all buttons off on mine and it didnt pop (I also died). So I went back and made it so there was 1 button for fire and did ch. 1 again (I also died). I got it the 2nd time.

I did it about 4 times dying a lot and using continues. Nothing.
Final time it worked when I died 3 times and used 0 continues. Seems like you can't use a continue.

How are you supposed to beat a boss without shooting? Does it eventually go away?

For the boss fight you have to survive long enough for the timer to reach zero, at which point the fight automatically ends.

I would down-vote this if I had a single achievement in the game, simply because you state that it is "easy". Some more info would be nice, or a video to show the best way to avoid some enemies. Is it easier to do in co-op? I cannot make it past the boss firing my weapons let alone firing nothing. Sure I need practice at this game but details would be nice.