Why are the flowers on my runner beans falling off?

could be birds , they can be a pest, try netting and see if this helps.

I was told some years ago that if the flowers don't 'set' they'll fall off &, of course, you won't get any beans.
Apparently in dry weather (ha!) you should spray the flowers with plain water which helps them to 'set'. I've been following this advice & it does seem to help :))

I would certainly "stop" the plants ...and keep on feeding them with tomato feed or similar liquid feed (LOW nitrogen though if your plants are healthy otherwise) so that more flowers develop. Spraying is fine but in a year like this...need I say more! It IS very important to water them well at their roots - they are thirsty things beans! I had a problem with flower drop and I grew nasturtiums up along with the beans....that brought more bees in to pollinate and kept the birds out too! Although I doubt if the birds were a problem with me...too well fed elsewhere in the garden! Too late for you this year of course but if you have any pots of flowers you could move near - it might just bring in the bees to pollinate the new flowers and so revitalise your crop.

Yes lots of watering at the roots is essential - ours are in deep pots (one of them is our old water cistern) & I've even watered them in the rain as I reasoned that all the foliage was stopping the rain reaching the roots (?) - also sprayed the flowers if it was windy. Both pots are immed. outside the house & there've been lots of bees buzzing round.
Don't let the probs stop you trying again :))