Every morning at work I have a cup of hot chocolate. I used to drink each one out of a disposable paper cup. Now I have a mug that I rewash each morning. On the surface, that seems more environmentally friendly… but it occurs to me that it takes a non-trivial amount of water to wash the mug each morning, plus some detergent, and paper is theoretically a renewable/recyclable resource.
So… which is more environmentally friendly?
You’d have to check with your local recycling authority but where I live, they don’t consider “food containers” recyclable, although paper cups are compostable. Perhaps it would be more environmentally friendly to wash the mug at the same time as other dishes? Or just rinse it out daily and bring it home once a week to put through the dishwasher?
Practically speaking, there are much bigger sources of waste water and garbage than your hot chocolate mug or cup. (Like are you opening single-serving packets of hot chocolate each morning? Would a tin of hot chocolate mix be less wasteful (and cheaper))
We went from cups to mugs to save money and be greener. Boy, did people ever get their panties all in a wad over that one. Sheesh.
There is a general trade-off between “disposable/land fill” environmental issues and “water use” environmental issues that flips depending on where you live.
Both manufacturing and recycling of paper products uses considerable water. Paper mills are among the top water-polluting industries, though using unbleached products mitigates that some.
there are people that think the way to deal with a coffee cup is to pour some fresh coffee into it.
if while the hot chocolate remains are still fresh you put some water (1/8 full) into it, swirl it and drink it, repeat until clean enough. you will then have a fairly clean mug, have wasted no water and had an extra mug of water. some might then only wash it once a week, some longer.
Most of these kinds of things depend on how much you value an apple versus an orange. But remember it takes a fair amount of water to make the paper to create the cup, and a fair amount of energy to make it and transport it to the office. It’s not just the trees that are costs of making the paper cup.
Washing the mug uses some water, and perhaps some energy to heat the water if you use hot water. Also some energy to make the mug in the first place, but I think that’s going to be pretty small in the scheme of things unless you’re breaking mugs every month.
But I’d guess the mug comes out ahead unless you’re really going to town with the hot washwater. If you really want to be green, and it’s your personal cup not shared, just quickly rinse the cup most days, and only do a thorough scrubbing once in a while.
Even if you disregard the water used in paper production (which is by itself probably more than that used in washing), water is also a renewable resource. The problem isn’t that it’s limited; it’s just that most of it isn’t in the places where we want to use it.
On the other hand, putting paper cups into a landfill is a carbon sink that helps to prevent global warming. 
(Darn, we really need a smiley just for playing devil’s advocate).
But, seriously, I’m all in favor of washing mugs. At my office, I take all the mugs home and run them through my dishwasher. Since I often have to work to fill up a dishwasher load at home, this is a perfect use of resources I’m going to spend anyway. In the meantime, I think providing mugs with our logo is a good image-booster for the company, and I got them cheap enough that I don’t even mind if clients keep them rather than leaving them for me to wash.
My city (Munich) has no lack of water, but decided several years ago to stock and rent a small wagon - the dishwasher mobile - which contains one industrial dishwasher plus an assortment of plates, cups and cutlery for your average neighborhood/ church festival, instead of paper cups. It’s required to use, and I think it’s easier to just collect the dirty dishes and run them through than collect all the garbage from throw-away plates and cups. I also assume that the city thought long and hard and spoke with experts and looked at figures before deciding that.
Until very recently, throwaway dishes and cups were made from plastic, not paper, which was a huge waste of energy and raw resource of petrol. I don’t have a lot of numbers for paper.
But if water for washing is very scarce because you live in the desert, and the paper cups are made elsewhere where water is plentiful and trucked in, the local decision for you might be different.