Voltage drop across LED in open circuit
It comes down to the LED still conducting current when it is not illuminating. Take a look at this characteristic from an OMRON LED. It's very rare to see stuff like this in data sheets: -



This shows the volt drop (about 1.3 volts) across an LED when 1 uA is flowing and it's not unreasonable to expect all LEDs to behave within a ball-park or two of this. So you are seeing a volt drop across the LED due to the current through the meter's inputs when placed across the open switch. If the meter's input resistance is 10 Mohm then there is about 0.345 uA flowing and there will be probably over a volt dropped across the LED should it be the device in the picture above.