It’s hard for anyone to hear the T because the next word begins in a D, which is very similar. In reality, we listen to the length of the preceding vowel. The vowel in “date” is pronounced significantly shorter than the one in “day.” This is not indicated in dictionaries, because it is a standard feature of English that we always pronounce a vowel shorter before an unvoiced consonant (p,k,t,c,sh,f, etc) than in any other position. The vowel will be medium length before a voiced consonant like “Dade” (a place in Florida). If the vowel is the last sound in the syllable, it will be very long. Native speakers are not consciously aware of this — they think they’re hearing the consonant, but that’s false. (Btw, I believe some Chinese dialects, like Cantonese, have something similar, called in English “checked tones.”) You can go to a dictionary and listen, paying attention to vowel length. (Press the red “pronunciation” button): http://www.macmillandictionary.com/dictionary/british/day http://www.macmillandictionary.com/dictionary/british/date_1

As an English teacher, I am sorry! If you are listening for a normal "t" sound in date, you will not hear it from almost any American. However, "day" and "date" do not sound exactly the same; I say the "t" in date as a glottal stop. It is hard to hear at first, but with more and more practice, you will hear it.

There is definitely a "t" sound in date. I know of no accent in English that doesn't pronounce the "t". That is the only way to differentiate between the two words. I believe the speaker was incorrect - makes it tough for those taking the exam!