I am a non-visa national (American, Canadian, Brazilian, etc) and do not need an entry clearance to visit the UK. Can I 'renew' my visitor visa by going to France or Ireland and returning on the same day?

The IDI tells us that there is no fixed interval required between successive visits to the UK, and this strategy is within the rules. But there is no guarantee that you will be readmitted. Also, the IDIs also tell us about the so-called "spent-leave rule", so this strategy should be used with care.