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According to this answer, a unicorn's pure blood carries a curse:
You have slain something pure and defenceless to save yourself and you will have but a half life, a cursed life, from the moment the blood touches your lips.’
Philosopher's Stone - page 188 - UK - chapter 15, The Forbidden Forest
Are unicorns the only creatures, in the HP universe, with blood so pure that it carries such a curse?
If you consider 'people' as creatures, and Lily's protection over Harry a 'curse', then I would say 'Yes'. If you disagree on the above two statements what I'll write next will be totally irrelevant to the question.
Lily's sacrifice granted Harry the protection, which prevented Voldemort to touch or harm him. Curse targeting Voldemort only, maybe, but still a curse nonetheless. Consider the very quote you have provided, emphasis mine:
You have slain something pure and defenceless to save yourself and you will have but a half life, a cursed life, from the moment the blood touches your lips.’
Sound too familiar ? The 'pure and defenceless' thing is Lily. The motives for the murder('save yourself') is to defy the prophecy and assure Voldemort's immortality. As a consequence for a human murder, we know that the murderer gets his soul shattered into pieces('half life'). The blood also played major role when Voldemort takes Harry's blood in GoF, and binds Harry to life with his own.
Considering your question:
'Does any creature's blood carry a curse in the same way a unicorn's does?'
Certainly. The two curses are both carried by blood, and are carried in very, very similar way.
No, not definitively in canon. Whether the question is about slaying an innocent creature, drinking a creatures blood, or doing both, we're given no examples in HP lore of any other similar creatures. That doesn't preclude the possibility of such creatures existing, but not having been discussed in any canon media.
Dealing specifically with blood, there are few creatures mentioned in-universe that have blood-related magic:
In these cases, the animal blood is know to be useful, with no mention about how it's acquired.
We also have other cases of magical creatures with body parts or fluid that is magical in nature:
In these cases, the magical element can be obtained without slaying the creature. Of note is that unicorns can provide parts of themselves, even unwillingly. So, this really solidifies the idea that we're dealing with a curse born of the murder of a creature that's pure and defenseless.
While Evdzkhan's answer is interesting, I don't believe it quite matches the requirements. First off, Lily's Protection is always referred to as Lily's Protection. It's never referred to as a curse, only a protective enchantment, and it has nothing to do with the victim's blood.
"But I knew too where Voldemort was weak. And so I made my decision. You would be protected by an ancient magic of which he knows, which he despises, and which he has always, therefore, underestimated — to his cost. I am speaking, of course, of the fact that your mother died to save you. She gave you a lingering protection he never expected, a protection that flows in your veins to this day."
So, the result of killing Lily did nothing to Voldemort except break his soul a little more (something Voldemort was already accustomed to). However, the killing did something to Harry and Harry's blood (and also relate's to Petunia's blood, so far as the Privet drive "home" protection).
Now, we can look at Harry and see if he matches the qualifications. Does his blood curse Voldemort when Voldemort ingests it? We have two other answers here that help clarify this.
We also see that Voldemort doesn't have to slay Harry in order to get his blood for the potion. Pettigrew merely cuts Harry to draw out blood for a potion. Harry lives. So Harry can be discounted on both tests: He doesn't cause a curse, and he doesn't need die for the effects to happen.
One of the key factors about unicorns is their supposed purity and defenselessness, as noted by Firenze. I would say they're not defenseless:
"... nearly gored me with his horn after I plucked his tail."
-Ollivander, recalling a unicorn attack during the retrieval of a unicorn tail hair used in Cedric Diggory's wand.
They're also described as incredibly fast, with a variety of magical perks. This leads me to believe Firenze's beliefs about the curse may not be entirely accurate, and are tinted by his close relationship to mother nature and philosophy. The slaying and blood drinking may be essential in bringing on the curse, but there's no proof that it's purity or inability to defend that triggers the curse. We've only the word of a Centaur on that. (I have nothing against Centaurs, it's just that aren't known for their willingness to cooperate with Wizards, and we're not given any insight into how his claim was more or less proven).
So, I would conclude that:
Unicorns are unique in regard to the curse upon their blood.
And I would conjecture that:
It's a magical self-defense mechanism to discourage killing them, rather than a curse due to their innocence.
There's also an open question as to whether or not unicorn blood that freely given or naturally happened upon would give no curse. A similar question was asked specifically about Harry giving his blood to Voldemort willingly, which shows us that the nature of this kind of blood magic, that is blood gained with a certain intent, is not clearly understood.
