It keeps showing Err. We've changed the battery and it still keeps showing that.

Did something get spilt in it or you may have a bad connection in the lamp area.check all connections and if lamp stays on when key isn't in ignition and keeps running it will over time kill the battery

Usual cause is bad shutter

Not sure if this will help any other owners since my answer is 4 years late, but this is what I found from another user. Apparently if you mess with the wiring too often, it goes in a safe mode which you need to do some button trickery to unlock the device from it's safe mode. These are the Steps.
1.Disconnect power supply
2.Press [MENU] and [UP arrow] buttons and keep holding.
3. While holding the buttons [MENU] / [UP arrow], connect the power plug again
A. The display will show "DC ERR"
*** if it still shows "Chk wiring ..." repeat the steps above until you get "DC ERR."
4. Press [UP arrow]; the display shows "DC1 ERR"
a. Press and hold [UP arrow] and it will show "DC1 OK"
5. Press [Down arrow], the screen will show "DC2"
a. Press and hold [DOWN arrow] and the display will show "DC2 0"
6. Disconnect the power from the back of the unit.
7. Remove the Face Unit.
8. Reconnect the power from the back.
9. Use a sharp object and press the [RESET] hole for 3 seconds and release.
10. Reinstall the face unit and turn on the power.

Usually I change the weight by weighing one foot and then weighing again with both. And then I keep repeating that until my results are consistent. Then that result seems to be accurate

The manufacturer has written this in their manual for your unit...

"Err" displayed -battery icon may show any state of battery capacity. One quick beep every 30 seconds. Unit is Error condition

Replace batteries. If "Err" (or blank display) continues, unit has malfunctioned. Replace immediately.

Link to manual...

http://portalgb.knowledgebase.net/utility/downloadArticle.aspx?aid=294720

You're right..This happens often when one changes the battery.Here's what to do :
The ERR message can only be corrected by holding in the DISP/MENU and STOP buttons for two seconds, which results in ALL the material on the recorder being erased. It will say (FOR) for a few minutes while it is erasing everything, then will work again.
Solution found at http://www.amazon.co.uk/Olympus-VN-2100-Digital-Recorder-microphone/dp/B000V3NTEU

Check Nikon for updates.

Answered here:

http://www.fixya.com/support/t809061-chk_error_screen_olympus_vn_4100_pc

turn key to <acc> turn radio will blink err. leave on for at least half hour.err will be replaced by enter code.