Должна быть is translated here as "has to be" but not "needs to be" (marked incorrect) or "must be" (also marked incorrect). So I need some help on context. Can I rule out a statement of incredulity, e.g., "The shirt must be here; I saw it here yesterday"? Can I rule out a statement of necessity, e.g., "The shirt needs to be here; I explained to the cleaner that I have a dress rehearsal this evening"? What is the context for "The shirt has to be here" that "must be" and "needs to be" do not fulfill? Thank you in advance.
Both "The shirt must be here; I saw it here yesterday" and "The shirt needs to be here; I explained to the cleaner that I have a dress rehearsal this evening" interpretations work just fine here.
"Must be" and "needs to be" are not incorrect, so they probably were not accepted for technical reasons. Try to report them if you encounter this sentence again.
"supposed to be" implies something like it "it should be, but it isn't".
The shirt is supposed to be here but I can't find it, so may be I put it elsewhere.
Whereas if I say "it must be here" or "it has to be here" then I am sure it is and I will keep looking for it until I find it.
It's not a technical term of grammar, but I think of this usage as a "conjectural mood" - expressing a lot of different concepts: the shirt is supposed to be there, should be there, was promised to be there, will ruin my plans if it's not there, could not be anywhere else, is where I remember leaving it, etc.