Firstly, it only sounds funny to people who know the joke - and they aren't likely to laugh at you for it... if someone misunderstands this sentence, it's because they have deliberately misunderstood it, for entertainment purposes.
The important thing to recognize is that it's a very harmless joke - there's no embarrassment involved, nor is it at anyone's expense, it's just silly. Anyone making the joke when you ask them to call you a taxi will certainly phone a taxi for you afterwards.
This type of wordplay is generally seen as clever and harmless, and most of it's charm comes from the fact it is completely unexpected by the person making the original (innocent) statement.
If you really want to avoid it, you can't change the sentence emphasis to emphasize/stress parts to avoid the ambiguity, because it's deliberate ambiguity. You can, however, change the "call" to another suitable word, or you can restructure it to avoid the phrase "call me a taxi"
Could you phone me a taxi, please?
Please could you call a taxi for me?
Can you get a taxi for me?
Please could you get me a taxi?
The important (well, unimportant) thing is to remove the "call me a" phrase, either by replacing "call", or by separating it from "me".
And as an aside, if you'd like to see more examples of this humour, the film Airplane is a great place to start. It's full of this kind of "using the language in an unexpected manner for humorous results"
Elaine Dickinson: You got a letter from headquarters this morning.
Ted: What is it?
Elaine: It's a big building where generals meet, but that's not important.
Ted: It's an entirely different kind of flying, altogether.
Rumack, Randy: [all together] It's an entirely different kind of flying.