The courier company who delivered my visa package to me damaged it. The envelope that I'm supposed to give to the U.S. immigration officer at the airport—it says "do not open"—is ripped. Can I go ahead and travel to the United States and just give the officer the ripped envelope?
When you got the visa package from the courier, your passport and visa were inside, along with an "immigration data summary" and the sealed envelope. If the only thing damaged was the courier's envelope—the big one holding everything else inside it—you would have no problem.
But you say there was a lot of damage, enough to also rip the sealed envelope that was inside. No matter who accidentally opens or damages that envelope, you have the same problem—it's not supposed to be opened!
The U.S. immigration officer at the border (an officer of Customs and Border Protection, or CBP) will be looking at the envelope closely. He or she might not let you into the U.S. if the envelope has been opened or damaged. If there is just a small tear, or so little damage that no one could possibly have tampered with the contents of the envelope, the officer might not notice, or might let you in anyway.
If you don't want to take the risk of being refused entry, the safest thing to do is to call the U.S. embassy or consulate where you got the visa, and soon. Ask the embassy or consulate to give you a new envelope in exchange for the damaged one. You won't need to send back your passport—just the envelope.
The embassy or consulate might allow you to bring the damaged envelope back and pick up a new envelope in person, or it might require you to send the damaged envelope back by courier. If you will not have enough time to get the envelope back by courier before you need to travel, you should ask the embassy or consulate to allow you to pick it up in person.