Nexus 5 multitask button not working after upgrade to Lollipop
I believe the issue has to do with the Tasker and/or Twilight apps.

I don't know why, but this is what solved my problem:

I rebooted into Safe Mode (press and hold the Power button until the "Power Off" text shows up), then hold that("Power Off" message displayed on the screen) down until it turns into "Reboot to Safe Mode".

After running in Safe Mode, I noticed that the Home and MultiTask buttons work correctly: ie. its very likely that an app is causing the issue.  I restarted the phone back to normal mode, and the buttons continued working.

I disabled Tasker and stopped Twilight, and the buttons began working as expected.  I've since re-enabled them both, and they both started working again perfectly, and the Home and Multi Task buttons work as well.

So either something was wrong with one of the apps, or just the act of turning on Safe Mode fixed something.