"What do you have under the shirt?"
Translation:Τι έχεις κάτω από το πουκάμισο;
Is 'από' always needed in a sentence like this? Or could you ask: 'Τι έχεις κάτω το πουκάμισο;'?
Yes, it's needed.
κάτω, πάνω, πίσω etc. are adverbs, not prepositions; they can't "rule over" a noun, so you have a form a compound preposition by adding από or σε.
In this case ("under"), it's κάτω από.
(Compare e.g. English "because" which can stand alone sometimes, e.g. "because it was raining", but can't stand before a noun as in "*because the rain" -- there, you need a little preposition "of" to turn it into a kind of compound preposition "because of" which can then stand before a noun: "because of the rain". Similarly you can Πηγαίνω κάτω "go down" but you can't then be *κάτω τη στέγη "*down the roof" but have to be κάτω από τη στέγη "under the roof".)
I added σου to the end of the sentence (it is wrong apparently). I think Greek uses possessives more often and I see it almost as a natural translation of "the" article in this case