the phrase: "Sit down" phonetically looks like [sɪt daʊn]. The "t" and "d" are in the same tongue position. Can we drop the "t" in the first word in this situation in fast/casual speech? like this: [sɪdaʊn]
You don't need to be talking fast. Something does happen, and you could call it "dropping", but /t/ isn't necessarily phonologically deleted. If you do delete /d/, you'd get flapping and the result would be the same as if you deleted /t/. Or the tongue gestures overlap in time, giving the same result as segment deletion. I don't think you can do that in British English.
EDIT: BTW, this is not a general process, it's limited to a few fixed expressions. Compare most instances of /t#d/ as in "straight down", "Put Don on the chair", "Hit Darlene", "sit downwind" and so on, where there is no dropping. It's the same lexicalized contraction process that you have in "will not" → "won't", "going to" → "gonna".