No, I don't think they do. I've had a phone contract for years and never been contacted about it.
I think it's up to you to act when it's up and either upgrade your mobile or switch to a better deal elsewhere.
More recently it's become the case that it's worthwhile to phone up say you want to cancel and then get offered a much better deal through 'customer retentions' as the cancellation department is now better known as.
