There is quite no case when it's not easier for the adversary. The only case is if the password is salted (with large random) and the salt is not displayed.
The main concern is to know how much easier it is for the adversary.
In the worst case, he can perform offline dictionary/bruteforce attack and due to the dispersion of the hashing function, he will have few false positives.
For example, if the website display 6 hex characters, you can assume that a bruteforce attack will statistically match a hash with the same heading 6 characters every 16^6 attempt (so one false positive every 16 millions attempt). It's a big advantage.
As previously mentioned, if the hash algorithm is bcrypt, scrypt, and so one, the speed of an offline attack will not be very interesting comparing to the online attack.
With abilities to perform an offline attack you can avoid account lockout due to passwords attempt threshold. So there is some cases where displaying part of the hash will permit to bypass an account lockout functionnality even if the offline attack speed is not huge (10 attemps per sec is still better than 10 attemps and then blocked).
It is also a way to stay quiet relatively to the SIEM and other intrusion detection mechanisms.
Note that having a huge randomly generated password will not be enough to ignore the risk. If the password is not salted and the hash algorithm is weak, a collide could occur with a shorter password attempt.