No. It can only fail one of two ways. If closed when should be open, dome light will come on. Or if open when should be closed, then other doors would still be able to work.
Here is the 2002 EWD (It is the same from '99 on. Please give your year):
http://tacoma.site40.net/4Runner_96-02_Repair&Wiring/techinfo.toyota.com/ileaf/ewdtmc2/ewdtmpdf/ewdsourc/2002/024runne/systemci/023il.pdf
The bad news is the 'brain' for this function is built into the gauge cluster circuitry (aka 'Combination Switch') and can't be replaced on its own. First, I'd check and see if the green wire with black stripe to the gauge cluster is getting ground when you open a door (pretty sure it's that color for all years.)
I can maybe give you some more ideas, but I need to know your year. If it is the cluster, the best fix might be to wire up a separate relay to control the dome light. Question: Does the Door Ajar light work when you open the tailgate? How about the other doors?