"Wear this shirt."
Translation:यह क़मीज़ पहनो।
I used to get this one right, until I learned (Rupert Snell) that ko after a direct object distinguishes, or individualizes, that thing: 'पानी पीयो' - drink water, 'पानी को पीयो' - drink the water. So I tried 'इस क़मीज़ को पहनो' and was smote by the duolingo judges. Am I off base here?
You are right and 'इस क़मीज़ को पहनो' should definitely be accepted.
However, the difference is not simply of definiteness or indefiniteness as the example seems to imply. You can say 'वह पानी पीयो' for 'Drink that water' (talking about a particular glass of water) or यह क़मीज़ पहनो for 'Wear this shirt' even when the object is definite. In fact, this is the most natural phrasing. What the को does is add a sort of 'weight' to the noun:
को is always used for sentient nouns. So, when talking about people, it's always आमिर को देखो, never आमिर देखो (unless we are talking about the movie called आमिर which I recommend btw). Similarly, when talking about pets or ascribing a personality to animals, को is always used. But you can say मैंने एक भीड़ देखी (I saw a crowd) because the 'crowd' is impersonal though it may be composed by people. I think this is what the book meant by 'individualising'.
For inanimate objects for which both forms are valid, the form without the को is the most natural phrasing but when को is used, you place some additional stress on the noun. This introduces a subtle change of meaning. For example, मैंने वह किताब देखी just means that I saw that book but in मैंने उस किताब को देखा, because of the stress placed on the object, it means that I saw that book with some deliberation. As a result, the second sentence is closer to 'I looked at the book'. With other verbs, this difference may not be as pronounced. For example, the only difference between यह क़मीज़ पहनो। and इस क़मीज़ को पहनो is that the latter is a more persuasive statement.