Almost certainly not.
It's not exactly clear-cut but I'd suggest that someone who drank unicorn blood without killing the unicorn would get neither the curse nor the benefits which Firenze is describing.
The full context of the passage makes this clearer.
"Harry Potter, do you know what unicorn blood is used for?"
"No," said Harry, startled by the odd question. "We've only used the horn and tail-hair in Potions."
"That is because it is a monstrous thing, to slay a unicorn," said Firenze. "Only one who has nothing to lose, and everything to gain, would commit such a crime. The blood of a unicorn will keep you alive, even if you are an inch from death, but at a terrible price. 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, Chapter 15, The Forbidden Forest).
Note that throughout this passage that Firenze is only referring overtly to slain unicorns. The prospect of a healthy, domesticated unicorn being 'farmed', as it were, for its blood is not one which is in prospect for Firenze. It seems that both the benefit (immortality) and the curse are tied to the act of slaying the unicorn. That's why the horn and the hair are used in Potions and wandcraft but not the blood. The horn and the hair can be extracted innocently from the carcass of a unicorn which has had a natural death. So can the blood, of course - but in that instance the blood is really just a messed-up cocktail with no true value. It's the slaying of the unicorn which makes the blood yield immortality. It has no other use, which is why it isn't used in Potions - and why people who forage amongst unicorn carcasses seemingly leave the blood alone. The blood is useless if the unicorn is alive or has died naturally.
"You have slain something pure and defenceless" - as Firenze describes it it's like the exchange of one life for another, an innocent creature's for a greedy human's. There has to be death, and intentional death, for it to work.
Of course, Firenze doesn't actually say "The blood of a slain unicorn will keep you alive...". He says, "The blood of a unicorn will keep you alive". Hence the question is a good one, and the slither of ambiguity about the rules on unicorn blood exists.
However, Firenze is only ever talking about slain unicorns in his speech. It would be weird if he were start talking about unicorns in general halfway through. I think we can assume that in order to achieve immortality by drinking unicorn blood you have to have killed the unicorn yourself, to slay it "to save yourself". Therefore, someone drinking the blood of a healthy, living unicorn would not receive the curse that Firenze describes. Neither would they receive immortality. They would simply be drinking unicorn blood for its own sake, which (unless it's your aperitif of choice) is a rather random and futile thing to do.