There aren't many disadvantages to having the eyes. Any disadvantages would be ones
caused by the player himself. Oftentimes, people with the invisibility tend to be
overly aggressive. This usually gets them into dangerous situations, like walking
into someone's grenades, etc. Another common mistake is that they feel like they
don't want to waste the eyes. This leads back to the over-aggressiveness, where
they feel they have to attack or it'd be a total waste. This should not be the
case. The eyes are a nice edge to have over your opponent, but they last for only
30 seconds -- In addition, you sustain the same amount of damage when hit and you
still have to hit your opponent the same amount of times in order to kill him. Yet
another common mistake people make is that they think they are invisible AND
quiet. You can still give away your position pretty easily if you pick up items
or fall off ledges. The biggest rule of thumb here is, do not overestimate the
advantage of being invisible.
Countering the Ring of Shadows
Wait for him to come to you
This is probably the best way to defend yourself against an opponent who is invisible. An enemy's eyes are a lot harder to see if you are moving. Thus, find a good spot where you can see all the entrances into the room you are in, and wait for him to come to you. Make sure there's no way of him sneaking up behind you. If possible, try to be at a higher ground than your opponent -- so even if you lose track of him, at least he can't barrage you with rockets.
Again, there is no shame in hiding. Of course, if it's a free-for-all, you can just go to the opposite side of the level and get kills. However, if it's a one-on-one, just hide in a safe place till you hear the eyes run out.
Pretend like you don't see him
If you're loaded and armored up, wait for him to come to you. However, when you see him coming, pretend like you don't see him. If you spot him from a far distance and fire a rocket at him, you may scare him off. Wait for him to walk closer, then suddenly pounce on him when he's least expecting it. He may think that you don't see him and will venture out into the open area. Blast him in the face and you'll have a couple rockets on him before he can get a shot off.
Use the lightning gun
When in a rocket fight with an enemy who has eyes, it can be close to impossible to track him for the entire length of the battle. As stated above, eyes are a lot harder to see when they are in the air. It's not uncommon that you bounce your opponent into the air when fighting with rockets -- that is why you should use the shaft. The shaft is a powerful weapon -- It will slow down your opponent, which will make the eyes easier to see, and it will also draw blood. When shafting someone, you will see red blood come out from his body. This is also the case with someone who is invisible. If you shaft them when they are invisible, you will still see the blood, which makes them even easier to see.