A circuit in this context is the same thing as a press tour.
From "How to Survive a Press Tour" at Entertainment Weekly:
By the time this issue of EW hits the streets, I should be done with an Around the World Tour promoting Larry Crowne, which opens July 1. The itinerary calls for 13 cities, 30 days, crossing all 24 time zones and the international date line. This sort of Celebrity Mule Train is familiar in every branch of showbiz: a grueling combination of political campaign, global race, exotic vacation, and vision quest marked by exhaustion, jet lag, and introductions to far more people than your brain can recall. Like most travel, there can be high points, and if you like the movie you are promoting (and I do), it doesn’t have to feel like the contractual obligation it is.
… You’ll be spending all day talking about the film to what will seem like a million people — hard-hitting journalists (who have done this a million times), crack publicity experts (who do this for a living), and room-service waiters who are astonished by how exhausted your face appears.
In short, somebody who can't do the circuit is somebody who, for one reason or another, is unable to partake in a press tour in order to promote their product—in this case, the movie they have just made.
The use of circuit comes from the following sense of the word:
3 a : a regular tour (as by a traveling judge or preacher) around an assigned district or territory
// The preacher ministered to each congregation in his circuit.
3 b : the route traveled
// His circuit took him to many towns in the county.