It's "do the dishes."
The Grammar Exchange hasn't found "do up the dishes" in any of its references; however, a creative speaker might add the particle "up" to broaden or intensify the meaning of a verb even though the expression has not (yet) become part of the English lexicon.
In your sentence, if you should add the particle "up," it would slightly change the idea: it might imply cleaning up a lot of dishes, more than the normal number, and perhaps not even directly after a meal, possibly the next day.
The phrasal verb "do up" has this meaning according to the American Heritage Dictionary*:
"1. Fasten up or put up; also, dress up. For example, Let's do up all the gifts in matching paper, or Please help me do up the collar button, or Jane did up her hair for the dance, or The children were all done up in their best outfits."
"Do up" can also refer to extensive work on a decorating project. The Collins COBUILD* includes this definition:
"If you do up an old building, you decorate and repair it so that it is in a better condition; used mainly in British English. Nicholas has bought a barn in Provence and is spending August doing it up. "
"Do up the dishes" might be more easily said, if at all, in British English than in American English.
As for "make the dishes," it doesn't happen. You make the beds and do the dishes.
You could "make a dish" referring to making a particular plate of food as in:
Grandma makes a wonderful dish every week when we go there for dinner.
*The American Heritage Dictionary of Idioms by Christine Ammer. 1997 Houghton Mifflin Company.
**The Collins COBUILD English Dictionary. HarperCollins. 1995