Most labs are drawn into a handful of generic negative-pressure test tube types which contain various blood stabilizers. These are then taken to a lab where the blood is mixed with different reagents for testing. Even if infectious materials were involved in testing (which is not the case), none of the testing materials are present in the tube used for the lab draw. In addition, blood only moves into the collection bottles because they typically use negative pressure to pull blood in.

The only way testing could transmit something like HIV is if a needle was re-used between patients, but most venipuncture needles have locking safety mechanisms to prevent re-use.