Apparently, when I pronounce the tones wrong on 莎士比亚 I've said something that my teacher refuses to explain to me. It should be Shāshìbǐyǎ. What am I doing wrong?
I think if you mispronounced 莎士比亚 (Shā shì bǐ yǎ) with a lighter tone, it could become something like 啥是逼呀 (Shà shì bī ya), where 逼 carries a meaning of female's genital which is quite rude. The whole sentence might turn out to be misheard as “啥是逼呀？” which means "What is a pxsxy(逼)?". Then it makes sense for your teacher not explaining that.
I don't want to explain it to you either, it'd lower the tone of this very serious Q&A site, but if you read as far as the second last paragraph of this http://languagelog.ldc.upenn.edu/nll/?p=15171 you'll get a very strong hint. The only thing I'd add to that is that 傻 is a very natural prefix, probably the single most common one, and 呀 is a natural suffix to almost everything.
It's hard to pinpoint for sure what you're saying and what your teacher is hearing, but we can make an educated guess.
莎士 is very easily confused with 吓死 for non-native speakers.
比 can easily become 屄, especially without proper tone mastery.
亚 could, like @steveLangsford pointed out, change to 呀 without much effort.
So now we've got something like the neologism 吓死宝宝了 (scared little baby me to death) but which you've said as 吓死屄呀 (scared (little) cunt me to death). This would explain your teachers discountenance and subsequent refusal of explanation.