Deliberate mistake: When the janitor pulls the plug on the phone, there would actually be nothing but dead air because the janitor just pulls the plug on the handset, not the whole phone. So they added a dial tone for comedic value. (00:22:20)
Continuity mistake: When Turk is holding the chili-dog, it changes from one handed to two handed and back between shots.
Continuity mistake: Elliot, Carla and JD are in the cafeteria, JD has an orange drink that's flat on the table with his hands around the base, the shot cuts to showing his front and the bottle is now tilted towards him with a hand around the top.
Continuity mistake: Right after Kelso thanks Cox and walks away, one earpiece of Cox's stethoscope is hooked under his coat lapel. When the shot changes to show Cox turning around, both earpieces are free and hanging lower on Cox's chest.
Answer: No, it was a fake, they used a stamp.
It is real.