Usually yes. But understanding the original pun will often be helpful in trying to come up with your own jokes that fit in there.

For example, if I didn't knew that “Comed-Tea” is a pun on “Comedy”, I wouldn't be able to come up with a suitable German name for the drink. Once I noticed that, I was able to think of a analogous-yet-different joke that works in German (which is a pun on the German word for “weird”, by the way, since that fits nicely.)

(OTOH, I probably don't have to worry about context when translating these one word-omakes…)