Will masking tape take off the paint that is already on the model?

You need to ensure that the model has had sufficient time for the paint to dry, but otherwise there should not be a problem. Modelling masking tape is designed to respect the model.

Most modellers masking tape has low tack which means that they will stick to the model but peel off very easily.

If you have to use household masking tape which is much stickier then I would suggest that you cut the tape to length and before putting it on the model stick the tape to your T shirt or other item of clothing and peel away, this will remove some of the stickiness from the tape.

Make sure there are no bits of fluff or hair on the tape and apply to model, this should produce better result and peel away easily.