I have deleted old spreadsheets from My Documents, but they remain listed in the open box on the menu page.
Are you referring to File > Recent Documents?
AFAIK there is no nice button to remove these. But for what it’s worth, the list itself is saved into libreoffice/4/user/registrymodifications.xcu file (inside libreoffice profile directory), entries look like this:
<item oor:path="/org.openoffice.Office.Histories/Histories/org.openoffice.Office.Histories:HistoryInfo['PickList']/ItemList">
<node oor:name="file:///home/mahfiaz/abstract.odt" oor:op="replace">
<prop oor:name="Filter" oor:op="fuse"><value>writer8</value></prop>
<prop oor:name="Password" oor:op="fuse"><value/></prop>
<prop oor:name="Title" oor:op="fuse"><value>abstract</value></prop>
</node></item>
As @mahfiaz indicates you could edit your registry and manually remove the entries. Your question (without further clarification) appears to be asking the same thing as this question i.e., use the History Master extension.