Ich muss noch das Geschirr abwaschen
Does this mean I must wash another dish or I must keep washing this dish or I must wash this dish again?
If the position of noch is changed, does it change the meaning?
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None of your suggested meanings is correct.
Instead, it expresses that the dishes aren't washed yet, and the speaker
is saying that washing the dishes has priority over something else. Eg.
Können wir endlich ins Kino gehen? Nein, ich muss noch das Geschirr abwaschen.
Can we go to the to the cinema already? No, I have to wash the dishes first.
The only other valid position of "noch"
Ich muss das Geschirr noch abwaschen
doesn't make a difference.
About your three meanings:
I must wash another dish
Ich muss noch ein Geschirrstück abwaschen. / Ich muss noch mehr Geschirr abwaschen.
I must wash this dish again
Ich muss das Geschirr noch einmal abwaschen.
I must keep washing this dish
Ich muss das Geschirr weiter abwaschen.
In your example there were various tasks to do in the kitchen. All tasks are done, but one task, the washing up, is still to do. This is expressed by "noch". In this example "noch" corresponds with English "still". But "noch" has other uses too.
In sentences like this the word »noch« has two very similar meanings:
It has to be done now, to make it possible that something with a higher priority can be done:
Ich muss noch zwei Dateien hochladen bevor ich heim gehen kann.
I have to upload two files before I can go home.
(Going home has a high priority, but you can't go home as long als those two file aren't uploaded)
Julia muss noch ihre Hausübung machen bevor sie mit dir spielen kann.
Julia has to finish her home exercise before she can play with you.
(For children playing always has a higher priority then doing exercises, but finishing the exercises is a condition to be able to play)
Ich muss noch das Geschirr abwaschen bevor ich mit dem Kuchenbacken anfangen kann.
I have to wash the dishes before I can begin to bake a cake.
(You want to bake a cake, but the place you need for it is occupied by dirty dishes)
There is something like a to-do-list (don't need to be a real existing list, just things you want or have to do), and on this list are items that have to be done sometimes in the future (not now, and not to make it possible to do other things; just because they are on the list)
Ich muss noch das Auto waschen.
I have to wash the car.
(Washing the car is something that you want to do in the next days because you think that i better should be washed soon)
Ich möchte noch Japanisch lernen.
I want to lern Japanese.
(You are thinking of learning this language for years, and you really want to do it sometimes, maybe in 5 or 10 years.)