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.