My boss doesn't like it when I start sentences with "as", and I'm not sure if it's actually a problem.
A case where I would start with "as" would be:
As your new account manager, it is my responsibility to . . .
Starting a sentence with as is not a problem, and never was. The Corpus of Contemporary American English has 150000+ cites for sentences starting with as, across all registers and contexts, including academic writing. Your boss is completely alone in this.
However, what your boss might actually be objecting to is the so-called dangling modifier. Prescriptive approach dictates that what you are really saying in that sentence is that the manager is not yourself, but the responsibility. However, from a descriptivist's standpoint, that is a perfectly natural, grammatical construction that is widely used and understood.