I'd say it depends on what else went in it. Straight coffee or tea probably only needs a rinse, provided the mug is in good shape (if the inner glaze is chipped, or the inner surface isn't glazed more frequent washing is probably a good idea*. Straight coffee may in fact be antibacterial; I know straight tea is (that's why at a lot of Chinese restaurants, the pour the leftover tea out onto the surface of the table before they begin to wipe it down.
On the other hand if you are adding milk, sweetener, or citrus juice more frequent washing is probably warranted, since they spoil easier. By extension, if the tea/coffee blend you use contains actual bits of dried fruit (a lot do nowadays) it is probably best to treat it as if it has sugar added (because technically it does)
My personal headache like this involves my tea. I've never had much question about washing my mug or cup, because I usually don't use one (I find hot drinks make me very sleepy, so I drink most of my tea lukewarm to iced, out of a glass) when I am at home.
The problem is with what I have to do to take it with me when I go out (say to the city) for that I use a thermos, or more accurately a Camelback (since that allows me to drink the tea without opening the bottle, while keeping it from leaking into my bag). That DOES get dishwashed after every use, except for the straws which get hand washed (because they melt in the machine).
The problem involves the top. That gets dishwashed too, but the design means that there is always a sort of reservoir in the rubber nipple and inner workings that the machine never gets to (even if you pull the nipple off before putting it in) Eventually black mold develops, and I have to replace the top (which usually means buying a whole new thermos, as replacement parts are hard to order on their own) I assume it a sort of design flaw (obviously, the people who designed camelbacks never considered anyone would put anything but water in them**) And most of the other sports bottles either break readily or leak constantly. Guess it's just one of those annoyances one has to live with.
*with tea this can actually be a sort of selling point. For really good chinese tea the tradtion is to use a Yixing clay teapot (which is unglazed) to brew it. Part of the selling point of this is that some of the tea is supposed to be absorbed and retained within the clay of the pot (for this reason, you are supposed to have a separate pot for every kind of tea you could brew****, even though the things cost a fortune.) Since these pots are often treasured for lives and passed down as heirlooms, 5 and 600 year old examples are not unknown, and the rumor is that some of those have absorbed so much you can make tea in them by simply adding water, no leaves necessary.)
** I know from personal experience one should NEVER try and put anything carbonated in a camelback. The nipple is firm enough to keep the drink from leaking, but the jostling as you move will cause pressure to build up in it and so, when you put the nipple in your mouth and press down to take a pull, the pressure will release all at one and a stream of carbonated liquid will hit you in the back of the throat.
*** some real aficionados say you need a new pot for each season of each tea, since, like fine wine the same tea will taste different from year to year.
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