Resistors when using multiple BME280?

I'm wondering.. When using multiple BME280 sensors to the same pins on a board, do I need to de-solder the pull-up resistors on one of the sensors or can I leave them in?

Some general explanation on how this work and what the limits are will help greatly!

You need only one set of pullup resistors on the I2C lines, and the combined resistance should not go below whatever limitation is imposed by the Arduino and all the attached devices. 4.7K Ohms total on each line is generally recommended.

So yes, you may need to remove resistors.

And all the sensors have to be connected to the GND on the board too? Why is this?

Yes, the Gnds need to be connected. You can make a pigtail to break out the Gnd from one Arduno to each the BME280 device. Common Gnd is needed so all signals are referenced to the same level.