When a wh-question asks about an object or a complement, a helping verb/modal verb follows the question word:
What did you say? Answer: I (subject) said (verb) nothing (object)
Where is my car? Answer: Your car (subject) is (verb) in the garage (complement)
Where can I sleep? Answer: You can sleep in my bedroom.
In your sentence, which day asks about the object of the verb think - a simple present verb that requires do or does in question:
Which day do you think bla bla? Answer: I think Friday is blah blah.
Bla bla has no effect on the structure of the question, and so bla bla must not be inverted. That means you can't invert should and I: Which day do you think should I ..? is incorrect. You can substitute blah bla with anything
Which day do you think he will come over? Answer: I think Friday is the day he will come over.
Which day do you think I should tell him? Answer: I (subject) think (verb) Friday (object) is the day you should tell him.
To see that I should tell him has no such effect, change the verb think into some other verb that needs no further complement:
Which day do you perform? Answer : I perform on Friday.
If you remove do you think, then you'll be asking about the object (or rather complement) of the verb should:
Which day should I tell him? Answer: You should tell him on Friday.
PS. In question, the verb goes before the subject
Which day should I?
in present/past simple tenses, there are no helping verbs and therefore we add the words do/does before the subject, in questions:
I respect my mom. Question : Who do you respect?
I think you should tell him on Friday. Question: Which day do you think I should tell him?
It's important to understand that Which day do you think I should tell him? and Which day should I tell him? are two different questions asking about two different things