I'm basing my answer on the assumption that a One-Time Password is used as a second factor, in addition to a traditional username/password combination. If this is not the case, and the One-Time Password is the only factor, then Gilles' Answer is certainly more applicable.
Most likely due to Cargo Cult Programming, which means blindly following patterns that have been observed elsewhere, without understanding the real meaning behind them.
A developer may see the "password" in "One-time password" and happily make it
<input type="password">. Afterall, that's what it's there for, right?
Is there a disadvantage?
Security-wise, no. Disclosing a one-time password to a third party (e.g. through shoulder surfing) is not as problematic, because the password loses validity after one use, or after a certain amount of time.
The only imaginable downside would be a lesser user experience, as a user might have trouble ensuring that what they have typed actually matches the password they received.