Quirrell doesn't have a very large window in which to drink the blood.

According to this he should have plenty of time:

"Is it possible to Transfigure a living subject into a target that is static, such as a coin - no, excuse me, I'm terribly sorry, let's just say a steel ball."

Professor McGonagall shook her head. "Mr. Potter, even inanimate objects undergo small internal changes over time. There would be no visible changes to your body afterwards, and for the first minute, you would notice nothing wrong. But in an hour you would be sick, and in a day you would be dead."

• hpmor ch15

I could see the drinker getting sick

From the transfiguration rules:

"I will never Transfigure anything that looks like food or anything else that goes inside a human body."

• hpmor ch15

This presumably means don't transfigure anything into food. However it could also be interpreted to mean, don't transfigure food into anything. I am somewhat disappointed in McGonagall for not catching that ambiguity.

Also Quirrell is not a recognized transfiguration authority:

"If I am not sure whether a Transfiguration is safe, I will not try it until I have asked Professor McGonagall or Professor Flitwick or Professor Snape or the Headmaster, who are the only recognised authorities on Transfiguration at Hogwarts. Asking another student is not acceptable, even if they say that they remember asking the same question."

"Even if the current Defence Professor at Hogwarts tells me that a Transfiguration is safe, and even if I see the Defence Professor do it and nothing bad seems to happen, I will not try it myself."

• hpmor ch15

However since Quirrells past is unknown (as far as Hogwarts is concerned) he could be one of the best transfigures in the world and he wouldn't be recognized as an authority. Also I don't see Quirrell neglecting something as useful and versatile as transfiguration, so I would expect him to know how dangerous eating formerly transfigured food is.

Transfiguration sickness isn't because things turn into poison. Your body goes into a transfigured state, minor changes occur, and when you come back from that state things are different. It'd be tiny things. Huge problems would cause you to die instantly, but little transcription errors would kill you in the timeframe described.

Eg, your veins wouldn't match up right. The DNA in your cells would be just a little bit off and you'd get spontaneous cancer in your entire body. Some small percent of neurotransmitters and hormones would be transformed into slightly different ones... etc. None of that would be contagious or even harmful to somebody consuming it. But to the animal itself it'd be devastating.

Also remember that once the transfiguration reverts and you're back to yourself, you're in a stable state. The only issue is that you're not back together perfectly. Quirrell would only get sick if he drank the blood while it was transfigured and then it changed form while inside of him.