Follow the microwave manufacturer's directions. All of the OTR microwave installation instructions I've ever seen require you to securely fasten the microwave to the wall by drilling into studs. Not only that but a good chunk of them that I've seen (and have installed a couple myself) require you to bolt a metal mounting bracket to the wall.
Doing this through tile comes with a serious set of challenges that include properly drilling all the way through the tile into appropriate studs (which you will have to find through the tile if you don't remember to mark the stud locations).
You also haven't touched on venting-- in some installations you may find yourself venting horizontally through the back of the microwave, in which case if you tile first, you will need to cut through the tile anyway and take into account the extra depth added by the thickness of the tile when you determine how long your vent duct needs to be.
In all, this means that if you tile first, you will be putting many holes in the tiles. Not only that but the mounting bracket for the MW will probably scratch and damage the tiles as well. Finally if you don't take great care in properly securing the microwave through to the structure of the wall, you will have a very unsafe installation.
For these reasons, and following the manufacturer's instructions, you usually do NOT put finish wall coverings behind an OTR microwave. Instead you should purchase sufficient extra tiles for your backsplash and then keep them in storage. If and when you need to replace the entire OTR MW with a different one, then you can add tile to cover any bare wall showing.