First, I would like to confirm the following questions:
1. Does the issue occur on certain user mailbox via OWA or each user mailbox?
2. Which OWA folder’s email cannot be deleted, Inbox, Sent Items or each folder?
3. Which version of operating system is installed on the client machine, Windows XP, Windows Vista or Windows 7?
At this stage, I suggest you temporarily disable firewall and anti-spam for a test. If the emails still cannot be removed via OWA, please refer to the following article and use isinteg tool to check and repair Information Store.
Description of the Isinteg utility