I was working on a pilot install of Configuration Manager 2012 and right before we were going into production we wanted to clean out all the test application Approval Requests from the system. I was able to do this by going directly to the database which is most likely not supported by Microsoft, but in this case it worked for this project.
First, you should always backup the Configuration Manager database before doing any sort of modifications directly. Next, I go into the SQL Management Studio and run the following SQL Commands against the CM_<SITECODE> database.
Delete from UserApplicationRequestHistory
Delete from UserAppModelSoftwareRequest
Delete from UserApplicationRequests
Once those are run, you will see a nice and clean Approval Request view.