Why does 닫혀요 sound like 다쳐요? Shouldn't it be pronounced as 다텨요 because of the ㅎ?
Nope, that's almost like a phonetic rule in Korean. When ㅌ is in final position and is followed by 이/히 or any of the diphtongs since they also include 이 because 여=이+어 for example, it is pronounced as ㅊ . Another example to this phenomenon is 같이. ( pronounced [가치] )