Is the phrase "thanks, but no thanks" appropriate in an email sent to a supervisor?

It sounds extremely rude, I am afraid. I would assume mitigating circumstances for a non-native speaker, but the "no thanks" permits "thanks" to be interpreted as substantive, and thus has a highly dismissive connotation which should never be used with your superior, and neither with a friend you would like to keep.

The connotation that shines through (at least in the UK) is a sarcastic "Yeah right, you think you do me a favour? In future, please spare me your ideas."

Given that "thanks, but no thanks" is often used as a mocking euphemism for "gee, that's a stupid idea, I will of course not do that" I would suggest not using it on your advisor.

What's wrong with just writing "thank you, but unfortunately I won't be able to make it for reasons A and B"?

It's not the use of a "slang" expression that's the issue.

Saying "thanks, but no thanks" implies that s/he was trying to trick you with the offer to which you're replying. The phrase is imbued with a bit of hostility and disdain.

So, don't use it. Also, the way your friend phrased your email sounds flippant. Turning down an offer from someone's advisor should be done with more of a justification. It wouldn't hurt to be polite and respectful, and write something like

Hello Professor Smith,

I want to thank you for the offer of XYZ; unfortunately, I've already made plans to [important activity that clearly should not be canceled to take the offer].

[Possible counter-suggestion regarding the offer, e.g. doing something after that period of unavailability.]

Regards,
John

Still pretty short.

Notes:

• The reason doesn't have to be super-specific, but the more vague it is, the more it's likely to sound like an excuse.
• If your friend dislikes his/her supervisor, or doesn't appreciate him/her, or the offer - that's double the reason to be polite and respectful of him/her when rejecting.