Logging entries into a website form
1) Starting a new thread seems like quite an overhead for this task
  but I don't see an alternative (other than using a 3rd party tool like
  jersey which will presumably do the same thing anyway ?)


I would use there some message processing tool. Send somewhere and than process what should be processed. But it's much more complicated. I would use ExecutorService which is provided by java-8. I think it's old school to create Thread manually for this case. 


  2) Should I be doing something more sophisticated than creating a
  local RestTemplate each time ?


You can use @Configuration FactoryBean where you can create one spring @Bean for RestTemplate and inject this instance into your class where is postLog method located.

@Configuration
public class BeansFactory {
 @Bean public RestTemplate restTemplate(){
       return new RestTemplate();
 }
}


and inject in spring @Component

@Autowire private RestTemplate restTemplate;



  3) Is it generally the right approach anyway (it feels over
  simplistic)


From my perspective doing async work for REST with Threads is little bit tricky. And open to few potential problems like - too many Threads, by default all threads with the same priority etc. 

I think that for some async tasks the best way is to have publisher-subscriber approach. You are doing something, then you want to run something in the background to avoid waiting for response and return response without already finised async call.

To avoid doing another call you can push somewhere this event and process event somewhere else which is not depended on the current application. 

Check also https://spring.io/guides/gs/messaging-reactor/