(BANK_AREA.contains(player)) {
Entity bankchest = closestObject(BANK_CHEST_ID);
 
if (bank.isOpen()) {
bank.depositAllExcept(STEEL_AXE);
 
} else {
if (bankchest != null) {
if (bankchest.isVisible()) {
bankchest.interact("Use");
sleep(random(700, 800));
} else {
client.moveCameraToEntity(bankchest);
}
}else{
} walk(DOOR_AREA);
Entity largedoor = closestObjectForName(LARGE_DOOR);
if (largedoor != null) {
if (largedoor.isVisible()) {
largedoor.interact("Open");
sleep(random(600, 900));
}
 
}
One reason why it wont do anything, you have after one "} else {//nothing inbetween so it has no action to do here so it will not do anything. }" 
I'll rewrite this code for you as well. 
 
    public int onLoop() throws InterruptedException {
        Area bankArea = new Area(2444, 3086, 2441, 3082);
        Area treeArea = new Area(2472, 3106, 2459, 3081);
        //Checks if your in bank area and inventory is full it will bank for you.
        if (client.getMyPlayer().isInArea(bankArea) && client.getInventory().isFull()) {
            RS2Object bankChest = closestObjectForName("Bank chest");
            if (client.getBank().isOpen()) {
                client.getBank().depositAllExcept("Steel axe");
                client.getBank().close();
            } else {
                if (bankChest != null && bankChest.exists()) {
                    if (bankChest.isVisible()) {
                        bankChest.interact("Use");
                        sleep(random(700, 800));
                    } else {
                        client.moveCameraToEntity(bankChest);
                    }
                }
            }
        }
        //Checks if your inventory isn't full and if your not a tree area it will walk you there.
        //If the closed door it blocking you it will open it and walk threw otherwise ignores.
        if (!client.getMyPlayer().isInArea(treeArea) && !client.getInventory().isFull()) {
            RS2Object largeDoor = closestObject(2466, 2469);
            walk(treeArea);
            if (largeDoor != null && largeDoor.exists()) {
                if (largeDoor.getPosition().distance(client.getMyPlayer().getPosition()) <= 4)
                    largeDoor.interact("Open");
                sleep(random(600, 900));
            }
        }
        //checks if in wcing area and area isn't full to woodcut
        if (client.getMyPlayer().isInArea(treeArea) && !client.getInventory().isFull()) {
            //Woodcutting method in here.
        }
        //will check if its not in bank area and inventory is full
        if (!client.getMyPlayer().isInArea(bankArea) && client.getInventory().isFull()) {
            //walking back to bank area here.
        }
        return 50;
    }
}
There, Nice clean method loop untested but should work fine.