If the Green Arrow starts to mess things up for the city because his tremor pops up at the wrong time, the entire team (and his family) could face the consequences.
The drug clearly wears off; if it wears off at an inopportune moment when Digg isn’t able to take another dose, lives could be lost.
Add on the FBI investigation courtesy of Samanda Watson, and Diggle’s life isn’t going to get any easier the longer he keeps this secret from his teammates.
After all, Diggle seems to have an arrangement about picking up the meds from his supplier, and that wouldn’t be the case if he didn’t have to regularly dose himself.