Neither the Roku device nor the Amazon Fire TV stick (or, indeed a Now TV stick, which also converts a 'non-smart' TV into a 'smart' one) requires any form of ongoing payment to use the free apps on it (such as BBC iPlayer, ITV Hub, YouTube, etc).
You obviously need to pay if you're going to be using subscription services, such as Netflix or Disney+. If you've got a Netflix Basic account, you can use multiple devices with it (e.g. your TV at home, your iPad when you're out, and so on) for just your regular fee but you can't stream to two different devices at once. (A Standard plan allows you to stream to two devices simultaneously and a Premium plan allows you to use up to four devices at the same time).