Question: I recently have had a discussion about my friend's tone in his email.

His supervisor asked him whether he wants to attend a summer school in May. He wanted to thank him for his invitation and also wanted to keep the mail short. As a result, that is what he came up with:

> 
> Hello Professor,
> 
> 
> I have plans during that period. Thanks, but no thanks.
> 
> 
> Regards,  
> 
>  John 
> 
> 
> 

He is getting along well with his supervisor, but I believe that the phrasing is a bit disrespectful nevertheless. It might be the case that I am not a native speaker. 

Am I exaggerating the situation and this is something can be overlooked?

Answer: 
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.
This question might be better on <https://english.stackexchange.com/>, but I think you already have a few excellent answers. Let me add my own two cents' worth.

"Thanks, but no thanks" often comes across as sarcastic and even contemptuous to the native speaker's ear.

A far better phrasing would be: 

"Hello Professor,

Summer school does sound very interesting. Unfortunately, I have already made plans for that period. I must therefore respectfully decline your thoughtful invitation. Thank you very much.

Regards,
John"

It may be a bit longer than your friend intended, but it's not unduly verbose.