As long as you have at least ½” drywall installed and sealed properly (no gaps, tape, compound) on the garage side of the wall the firewall integrity remains intact (unless there are different requirements in your area like thicker or double drywall). However, removing insulation will make this section of the wall very cold during the winter (unless you have a heated garage).
Since the refrigerator produces significant amounts of heat at its bottom, rear, and / or top section (depending on the model) colder garage wall and hot air from the fridge could result in condensation and mold development. Depending how many inches you have to spare you could sandwich a solid foam board with aluminum facing towards the kitchen between the drywall and the plywood board.
Another option is to put the framing members (2×4’s or 2×6’s or whatever you have there) sideways which would give you only half of the wall’s depth (with 2×4’s) but you’d gain extra space for insulating the wall.
There’s also an issue of the wall framing. If you remove any portion of the wall framing members you should install an adequate size header that will be able to support whatever is above this area. You not supposed to just cut the framing, insert some spacers into the open wall cavities and finish the surface. You have to cut the wall higher and wider than the actual opening for the refrigerator in order to install the header and its size would depend on the opening’s span.
Unless you want to extend the fridge’s cavity into the garage, the above are probably your best options.