When we store our headlamps, currently we store them with the batteries. Should they instead be stored separate from the batteries?
I would highly recommend storing all gear with batteries removed.
Bonus Protip: Battery acid can be removed with a solution of baking soda and water. Soak the encrusted item in this (with battery removed) then allow to dry thoroughly before re-applying electricity. (Note -- might not be a good idea with higher-tech devices.)
There are two possible reasons not to store batteries in a headlamp that won't be used for a while: Residual current draw, and batteries leaking and destroying the headlamp.
To make things waterproof, some headlamps don't have a regular mechanical switch to turn them on and off during normal operation. For example, mine draws a few µA when "off". That's still very little for a pair of AA batteries, but when I'm storing the headlamp without a specific use expected within a few days I turn it off completely. On my headlamp, that means popping the watertight cover and moving a jumper. In that case it is truly off and the issue of residual current is eliminated. I don't know if all headlamps work that way.
Depending on how the on/off switch works, you may also want to remove the batteries just so that they don't get killed if the unit is accidentally turned on while it's in storage. If you keep it in the car, for example, consider that it could get jostled about. I was reminded the hard way last summer to do this with the camera battery when packed in luggage on a airplane. Somehow the camera had gotten turned on in transit, and the battery was dead when I went to use it. Fortunately it was rechargeable and I brought two spares, but it's still something I won't do again.
Old batteries leaking and destroying equipment isn't the problem it used to be. The old carbon-zinc cells were notorious for this, but that technology hasn't been used in a long time. Nowadays, this is not likely to be a issue.
So all that is a long way of saying, no, I wouldn't bother remove batteries from a headlamp that is being stored someplace where it won't be molested until the next use.
The only time I remove the batteries from my headlamp is when I'm bringing it as luggage on a airplane. I made my LED headlamp myself. A few years ago I was flying to Arizona using my backpack as checked luggage. A short while after I got to the terminal waiting to board the plane, I heard my name being announced over the PA system, requesting me to go to the ticket counter. There were two TSA agents waiting for me when I got there. They asked what I had in my backpack. I started listing stuff, like camera equipment, and when I got to the LED headlight they both said "that's it". I told them I was a electrical engineer and had made it myself. They actually found that rather amusing and said I was fine. As one of the agents was escorting me back to the terminal, I asked what I should have done to make it look less suspicious. He said I hadn't actually done anything wrong and they were just checking. Then I asked if it would have been better if I had taken the batteries out and put them in a side pocket somewhere. He thought about it and said, yes, that would probably have raised less suspicion. That's what I've done from then on and I haven't had another "encounter" with the TSA since.