Am I working harder to climb up a moving escalator than I would be if the escalator were still (assuming same speed of walking for both)?
Is the escalator moving up or down?
It efffectively changes the distance you walk. If you’re moving with the escalator you are walking a shorter distance; if you are moving against it you are moving a longer distance.
It’s a treadmill, basically.
Assuming you mean climbing an escalator going up, once you get up to speed and are no longer accelerating, you should be working an equal amount per stair climbed. Of course, you have to climb fewer stairs if the escalator is going in your direction.
At the beginning, when you’re getting up to speed, you’re probably working harder for a couple of steps.
(This analysis leaves off any relativistic effects due to climbing out of a gravity well – I have no idea if that makes it easier or harder, or neither.)
Sorry, I meant walking up an up-going escalator after initial acceleration. That is, once you’re underway, is each step “harder” because of the (constant) movement of the escalator or is it the same as if it were stationary.
Jesus, why was this question so hard to phrase?
I haven’t taken physics since college, but since you’re not accelerating, it should be the same whether the escalator is moving or still.
you’re facing (slightly) higher wind resistance when you’re walking up an “up” escalator. Each step should require (slightly) more effort than walking up a stationary escalator.
That’s an excellent point. The OP didn’t specify whether he’s climbing the escalator in a vacuum.
Are there any (tiny) relativistic effects relating to climbing out of a gravity well at a higher or lower constant speed?
Would you weigh more standing on an escalator that’s going up? I suppose not, after the initial acceleration when you step on.
There is a tiny [read: negligible] increase in wind resistance due to the slightly higher velocity.
The change in gravity from change in altitude will be the same overall. Unless your legs are 50 ft long*, there should be no difference in the gravity change between steps.
You will have to step fewer times on the escalator going up than on a still set of steps (nonmoving escalator to ensure the steps are the same size and all that).
As you step onto the escalator, your foot will be trying to move out from under you/move upwards, which will affect your balance. Most people can successfully handle this, since escalators are designed to keep this difference minimal.
You are not pushing yourself higher with each step on the moving escalator than you would be on the still one, just that the lower step is pushing you up at the same time. Ergo, you are not working harder, the escalator is working for you.
*Really bad WAG. Take with grain of salt.
Oh God, no. Don’t go there. 
Now that’s an episode of Mythbusters I’d like to see.
The important question is, can an airplane take off from a moving escalator?
Yeah, I went there. 
Each setp climbed would take approx. the same work moving or not moving. But moving it would less steps climbed to reach the top.
It should be clear that once you’ve stepped on, you have the range of options from standing still (which costs you nothing) to taking the stairs two at a time (which might get you puffing). You must subtract the escalator’s input from any work you choose to do.