Depends on whether you want to try growing some from seed. If you do then leave the flowers alone and collect the seeds when ripe and sow them straight away. If not then by all means remove the old flowers.

I would recommend removing the seed heads as hellebores are totally promiscuous so any seed that sets is going to be a hybrid - great if you have the space...

As with practically all plants, removing the nascent seedpods causes the plant to put more of its energy into its own growth, and next year's flowers.

Thank you to all of you - so, what I'll do is take off the seed heads from the plants in most of the pots but leave a couple to seed - what do I do then ? How does the seed spread - do the heads burst or do I collect the seed ?The plants are on the patio so if they do it themselves it probably wont land on any soil. Then do I put seed in pots or just let it take its chance in the border? You'll be able to tell I've only had hellebores since last year! Aren't they lovely

So keep an eye on the seed pods, as soon as they begin to change from light green to a darker shade they are getting ready to split. The pods will open easily when the seeds are ripe, black and shiny.
Sow them in ordinary potting compost, I prefer a deeper pot than normal as they have long roots. Cover with about a cm of compost or grit which ever you prefer. Put the pot somewhere cool and safe from rodents and leave until next Spring. Just make sure they do not dry out. Prick out when they have their true leaves and wait for a few years to see what you have got.
OR. Sprinkle the fresh seeds in the border where you want the plants and let them get on with it.

Thank you , that's really helpful