Early in the film, as Dr. Deemer goes into the lab and puts on his apron, he wears a shirt with short sleeves, but when he puts his arms in the isolation box, the sleeves are long, like a lab coat, then short again in the wide shot when he pulls his arms out.
When Paul falls to the ground and dies after attacking Prof. Deemer, the index fingers of each hand are extended straight out while his other fingers are curled inward. When Prof. Deemer comes to Paul's body later, all fingers of Paul's hands are extended straight out.
When the Tarantula attacks Prof. Deemers house, we see a closeup of its "face" looking through the window, with two large eyes right above the jaws. A real tarantula looks completely different, with eight eyes located at very different positions.
Spiders cannot grow without shedding. In order to get so large Tarantulas would have had to shed her skin many times, leaving a trail of empty exoskeletons behind. It would have also taken her a lot of time since every shedding requires to regenerate a new exoskeleton.
At the tarantula's final size, humans would no longer be a suitable prey for it since more energy would be expended catching and eating such a (relatively) small food source than would be gained from ingesting it.
Prof Deemer takes special care to fill the hypodermic needle inside an isolation box wearing rubber gloves; however, once filled, he pulls it out of the box with his bare hands and then removes air from the needle by shooting a little bit of serum out. Had the toxicity of the serum been that dangerous, he probably would have done that with the needle still in the box and would have worn gloves while handling the needle outside the box.
When Prof Deemer inspects the animals in his cages, he changes the card (on the clipboard) showing the age of the first animal (white rat) from 5 to 6 days and writes that the rat received the second injection on the 6th day; however, even though he holds the needle in his hand, he does not give the rat an injection. He then inspects the clipboards for the remaining animals yet does not change the date of the age for any of them.