I don't know why your workplace would say that. HIV is a fragile virus, and also, must enter the bloodstream in order to infect the patient. Any exposure to the environment causes the virus to start to degrade, and a lancet isn't going to inject into your vein. No one, even in a healthcare setting, has been infected by a lancet.
Assuming you are a healthcare worker, you should follow your institution's infection control protocol, but you shouldn't worry about this at all.