eye08074430-san, エリックsan,
kon'nichiwa!
eye08074430-san, do you mean people drop "h" like "h" in "herb"?
If so, it's not quite same. Like エリックsan said, it sounds "almost" like い, but not really い
It's more similar to snakes' hissing sound
I'd say, if there's any letter to drop, it'd be "i" in "hi", not "h": becoming "ohsashburi desu".
Because "h" is what's called "unvoiced" consonant, it's easily dropped and even vowel "i" becomes voiceless.
As a result, some people find this sound more like hissing sound without vowel, and other people hear
voiceless [i] without consonant.
Natsuko(奈津子),
Team JapanesePod101.com