Yes it is quite true.
The episode in question was filmed on 8th april 1986 which included John Lowe as the charity thrower.
A man called Bill Bridge, aged 59 and a heavy smoker was throwing at a normal dart board (not the bully prize board). As he was actually pulling his first throw of darts from the board he become faint and collapsed on to the board. He fell to he knees whereby jim Bowen and a cameraman went to his aid.
They realised he was havinbg a heart attack and sat him against the studi wall underneath the dart board. Whilst awaiting proper first aid, the dart board fell straight down, smashing in to poor Bills head.
Luckily he survived the heart attack but suffered from brain damage until his still early death in 1993.
He was born in Northamptonshire and left a wife, two children and a grand-daughter behind.