In the movie Constantine said, "You don't need its protection, it would be like the old days."
Did the amulet prohibit hearing voices? So he could find out what's happening and report back to Constantine or something else?
While neither the movie or the script specifically discuss what powers the amulet possesses or grants, John gives it to Angela as a ward of sorts. From the script:
John turns to her -- locks on those eyes. He reaches into his coat pocket as he walks up to her. She shudders as his arms wrap around her face and for a beat they're cheek to cheek. When he pulls back she sees he has clipped Hennessey's AMULET around her neck. JOHN Think of it as a bulletproof vest.
Here is where the script and the movie diverge.
In the movie, Angela accidentally leaves the amulet in the car when she takes off her jacket and it's when John notices the missing amulet that Angela is abducted.
So, added to John's earlier comment to Hennessey, it's reasonable to assume that it provides protection from attacks by demons/half-breeds.
As for exactly why he took - or, as in the script, removed the amulet - let's look at the script again.
HENNESSEY Oh God, you want me to surf the ether. He instinctively touches an AMULET around his neck. Four intersecting crosses. JOHN Come on, you know that exorcism wasn't right. HENNESSEY I... I don't have the Sight anymore. JOHN Don't have it or don't want to use it? Hennessey vacillates. This is obviously tough for him. JOHN Just look around. A few days. You spot anything unusual, anything -- you let me know. Okay? John wraps an arm around his shoulder like a good buddy, then reaches behind his neck -- JOHN It'll be like old times. -- and unclips the amulet from Hennessey's neck. That unnerves the big guy. John drops it in Hennessey's pocket. JOHN Just for a few days.
This backs up the idea that the amulet is a ward. By removing it, John hopes that Demons may come closer to Hennessey and he may be able to learn something.
Finally, we have the script version of the scene where John and Angela find Hennessey in the store.
John leans down to the remains of his friend. A cop starts to intervene but Angela blocks him. Let him be. John has trouble taking this in, reluctantly reaches into Hennessey's coat pocket -- finds the protective amulet he himself removed from his friend. That's even more crushing.
This final bit pretty much confirms (IMO, at least) that the amulet was meant to protect the wearer from demon attacks.
"His" protection in the whole movie (unless there is a back story to it) refers to God. Constantine displays this rebellious attitude the whole time, even as he was going to heaven for sacrificing his own life (he was showing the middle finger to Lucifer). The priest was psychic and such ability comes with a price. While the priest was holy enough he in normal terms would be open to demonic attack and makes him susceptible to being possessed or damaged in some way. Thus the amulet prevents such dark forces to intercept in between. However, in order to use that power he has to take off the amulet so that he is "open" to his normal "channels". Sort of like taking lithium for schizophrenics who hear things. This is spiritual lithium.
It's a holy protection trinket. It protects against evil. I'm assuming the voices the priest hears are all evil. That's how he finds possessed individuals and demonic possession. I also believe that's why John gave the trinket to Angela. To protect her from the darkness. This is just my understanding of the movie, I could be wrong. I also know that this flick was heavily criticized, although I thought it was awesome.