Nope, you don't have to fast...
But here's some information that is important if your doctor hasn't already mentioned it;
Because the blood level of lithium rises rapidly for a few hours after swallowing a lithium pill and then slowly levels off, having a blood test right after taking the drug can mislead the doctor into thinking that the dose is too high. To gauge the average blood level accurately, it is important to have blood drawn about 12 hours after the last dose of lithium.
Most patients take their nighttime dose of lithium and then come to the doctor's office the next morning to have a blood test before taking their first dose for the day. Some patients are able to take their full daily dose at bedtime and don't have to worry about the morning dose when getting a blood level.