Thanks for your help, I'm just getting started with level design, and all suggestions are appreciated.

public Transform assignThisInTheEditorItGetsDestroyed;
void OnTriggerEnter(Collider other)
{
    Destroy(assignThisInTeEditorItGetsDestroyed);
    Destroy(gameObject);
}

You put that script on a little trigger volume (collider with 'isTrigger' set to true), and then link it to a door object in the inspector. Then, if something enters the trigger, it'll delete the door, and then itself.

OR to do Zelda style one-use-antimatter-keys-

On your player (somewhere in your player script)

int keys = 0;

public void AddKey()
{
    keys++;
}

public bool CanOpenDoor()
{
    if(keys > 0)
    {
        keys--;
        return true;
    }
    return false;
}

Then, on your pickup a slight variant on the first one-

void OnTriggerEnter(Collider other)
{
    (name of your player script) player = other.GetComponent<Put the name of your player script here>();
    if(player)
    {
         player.AddKey();
         Destroy(gameObject);
    }
}

That gives you a key, and destroys the pickup.

Then on your door, something else similar (have a trigger volume along with an actual door- so that if the player enters the trigger with a key, the door opens)

public Transform thisIsTheDoorAssignItInInspector;
void OnTriggerEnter(Collider other)
{
    (name of your player script) player = other.GetComponent<Put the name of your player script here>();
    if(player)
    {
         if(player.CanOpenDoor())
         {
             Destroy(thisIsTheDoorAssignItInInspector);
             Destroy(this);
         }
    }
}

That's a little more sophisticated!

If you wanted to check for specific keys, you could go even further and do something like this-

[System.Serializable]
public enum KeyType
{
    Red,
    Blue,
    Yellow,
    Master
}

public int[] keys = new int[];
public void AddKey(KeyType type)
{
    keys[(int)type]++;
}

public bool CanOpenDoor(KeyType type)
{
    if(keys[(int)type] > 0)
    {
         keys[(int)type]--;
         return true;
    } else if (keys[(int)KeyType.Master] > 0)
    {
        keys[(int)KeyType.Master]--
        return true;
    }
    return false;
}

Then in your triggers for the pickups and the doors, use-

public KeyType key; // Assign this in the editor! 
//blah blah then in the trigger bit
player.AddKey(key);

// and then in the door;

if(player.HasKey(key))
{
    //Do things
}

(Thanks, @wccrawford, for the suggestion!)

Then you'll need a few scripts.

When the player collides with the 'object', the script needs to set a variable (probably on the player) that says the player has the object.

When the player activates the door, the door's script needs to determine if the player has the object.