Hi Cornelius,
You need to check the terms and conditions on the newspaper's website. In all probability the individuals who post will retain copyright in their entries, but will have granted the newspaper forum a non-exclusive licence to publish and archive their work. There is an outside possibility that the newspaper has done a 'rights grab' and will claim that the posters have, by signing up to the service, agreed to assign all copyright to the paper.
Obviously you will have great difficulty in contacting the individual posters for permission to quote their postings, but it could well be that if you only want to quote small chunk of their comments, this would be covered under section
30 of the Copyright Designs and Patents Act 1988, so long as you give an attribution for the quote.
The majority of blogs and tweets which re-cycle other people's news and comments rely on this exemption.
The formal and expensive alternative would be to get a website republishing licence from the
Newspaper Licencing Agency, but frankly I do not think this is necessary if you only use small snippets.
Afternote. If the site you were referring to is the Times Educational Supplement, then as this is behind a paywall, you also need to check what the t&cs say about what you, as a subscriber, may do with regard to content. Annoyingly, their site does not put their t&cs outside the paywall so I can't check this.