What causes a voltage drop at the input of this TL082CP Op-Amp?
Is your multimeter only 1 Megohm input impedance? I would expect to see at least 4V after R4 with a 10Megohm DMM.

As it is, you have a potential divider between 4.5V and VIN+, formed by R4 and the DMM itself.

When you move the DMM to make any other measurement, the potential divider goes away.

I think this is enough to explain all your observations.