Remove last year's cotoneaster berries ??

[Sae Nguyen]

Or leave them on?

 

[penumbra]

They will fall off when new growth starts. But you can take them off if you want to.

 

[Shibui]

Berries don't last that long here unless I protect them from the birds.
I doubt it will worry the tree whether the berries stay or go so that leaves it up to you whether to remove or not.