Your problem here is the counter top, as moving it may pull the tiles away. You could just try and remove the grout between the counter top and the bottom tile and then remove the counter top and units. Once done you would need to replace the grout.
The issue here is that often the base units get screwed into a batton on the wall and this screw is in the space at the back of the unit and may not be accessable from the inside of the unit.
If it were me I'd unscrew the counter from the unit you want to remove (screws go up thru the inside top of the unit) and also the screws which connect the units on either side. If you're lucky and the unit in question is not screwed to the wall it may slide out. You may need to adjust the feet down a little, if they are adjustable, to make it come out a little easier. In this way you would not need to remove the counter top.