I was thinking about this more.
A cup of tea is a bit less than 3 cents. The cost of water from the tap isn't enough to be worth calculating (and in a world of shortages, I live in a region where we have too much water). Heating is about 1/4 of the total cost, which is significant.
But that leaves me with a dirty cup that eventually needs to be washed.
So how much does it cost to clean the cup?
Starting with my kitchen faucet that had been doing nothing for hours, I started a clock and turned on the hot water and measured the volume that came out before it became warm-enough-to-wash-some-stuff. It took about 18 seconds and 1.6l to get to right about 110f, which my fingers determined to be "warm enough". The incoming water temperature is 58f right now. (The water heater is a recently-installed resistive unit.)
Raising the temperature of this 1.6l of water cost me almost exactly 1 cent, and that's all waste energy -- that's what it takes to get the process started.
Then I have to wash the cup. I didn't work through that, but I've done it enough that I can estimate that it takes me about 20 seconds total to scrub tea stains from a coffee cup and make it white again if I'm in a hurry about it. No big deal, time-wise.
But because I'm focused on washing the cup more than I am on conserving energy, I usually let the water run while I wash up the cup. That's another 20 seconds of running the hot tap (for 38 seconds total), adding another 1.13 cents (or 2.13 cents total).
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So that's our minimum: 2.13 cents, just to heat the water to wash the cup with 38 seconds of running it. Maybe, some days.
But realistically, it's likely more than that. I've measured incoming water temperature to be as low as 38f here[1]. The water heater itself is set to 140f, and as a practical matter when I wash a single coffee cup I generally use 100% hot water.
In that worst case, it's more like 4.23 cents to wash a cup. (Again, if I'm hurrying. I don't always hurry.)
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So the total cost of a single cup of tea, for me, is somewhere in the realm of between 5 and 7 cents. Heating the water is by far the biggest contribution to that cost.
And merely washing the cup by hand is a significant cost contribution at one end of the scale, and is the largest contribution at the other end.
That's surprising to me. These numbers still don't matter much (how many cups of tea do I drink in a month? or in a year? how often do I actually wash that cup?), but it's surprising enough that I think I'll start using the dishwasher for these things.
The dishwasher is a bit cheaper, and a lot lazier.
[1]: I'm not redoing worst-case for making the tea itself, because the water for tea-making comes from a large countertop water filter that connects to the kitchen faucet. The contents are generally always near room temperature, at the behest of the ambient environment that is controlled by a gas forced air furnace or aircon. I'm not getting that far into the weeds. :)