When I buy books, if it's a really popular title I usually try to get both the English and Japanese versions. For instance, I have the Harry Potter and Chronicles of Narnia full sets, both in Japanese and English. When I read the English I like to look out for things that might be hard to translate and then go to the Japanese version to see how it was handled.
In Philosopher's Stone, Harry asks Hagrid "What's the difference between a stalagmite and a stalactite?" And Hagrid says "Stalagmite has an 'M' in it." In Japanese, "stalagmite" is ?? (sekijun) and "stalactite" is ??? (shounyuuseki). When I read that I immediately pulled down my Japanese version from the shelf. When Harry asks "what's the difference between ?? and ????" (but in Japanese, obviously), Hagrid says something like "?? has two characters in it."
I also have The da Vinci Code in English and Japanese, and that one is full of poems, clues and word structures that only work in English. The Japanese one handles it surprisingly well, but some of the original meaning is lost to the Japanese reader. It can't be avoided.
Luckily I've never had to interpret Japanese puns on the spot in a professional setting. I interpreted at my friends' Japanese wedding once (American groom and Japanese bride). The families from both sides were there and some of them wanted to give speeches. The Japanese speeches were a little easier to interpret into English because they tend not to joke around when giving speeches, even at weddings. I was lucky that the Americans were pretty straightforward in what they wanted to say, but one guy said a funny remark that was meant to be an inside joke, so I didn't get it. Some of the Americans laughed, and when the Japanese looked at me for an interpretation I said "I have no idea what the hell he's talking about (???????????????????????????)" and they all laughed too. I cheated, but it worked!