London Pass

No, it does not. It says you get fast-track entry at “certain selected attractions” but the only ones specifically mentioned are the London Zoo, St. Paul’s cathedral, and Tower Bridge.

https://www.londonpass.com/how-it-works/what-you-get-with-the-london-pass.html

They make it very difficult to find information on others where it may be offered.

These are the only venues where the London Pass gives you Fast Track entry:

Tower Bridge [not the Tower of London]
St. Paul's Cathedral
London Zoo
London Bridge Experience
Kew Gardens
Hampton Court Palace
Shakespeare’s Globe
Cutty Sark

This "fast track" entry doesn't allow you to skip security lines. The information is on the page I linked, but if it doesn't appear when you first land on the page, click on the "See included attractions" link and the list will appear. (This is the first "included attractions" link on that page. There's a second link, under "Avoid the Crowds," which is apparently a list of venues which are "off the beaten track" to the extent that you shouldn't encounter any lines. You don't get to skip lines at those, if there are any.)

Thank you!

London pass holders do have fast entry at the Tower of London.