Does plywood over rigid foam meet the fire barrier requirements?
There are 3 considerations: 1) Fire Walls, 2) Flame Spread Index, and 3) Smoke Development Index. 

1) Fire walls prohibit (or slow) the passage of fire from one room to another. Fire walls are used to contain the fire to allow a.) occupants to escape, and b.) contain the fire to protect the structure. In residential construction the only fire walls are between a garage and living spaces. That is not your issue. 

2) Flame spread is based on a materials ability to burn and spread through a building. All building materials have a Flame Spread Rating and all buildings are rated for the type of Flame Spread allowed within the building. 

Residential buildings allow a “Class C” Flame Spread Rating. (See ICC Code Section 803 and Table 803.5.) A “Class C” Flame Spread Rating is between 76-200. (Just to give you an idea of ratings, oak has a rating of 100.) 

So, if your rigid insulation has a rating of more than 200, it needs to be “protected”. Protected means covered. Drywall, plywood, etc. qualifies as an allowed protection. 

3) Smoke Development Index is similar to Flame Spread Rating and requires a “Class C” Smoke Development Index too, which is between 0-450. 

Again, if your rigid insulation is rated above 450, then it needs to be protected AND drywall and plywood qualifies as protection. 

Summary:

Yes, plywood is allowed to protect the rigid insulation. Any thickness is allowed, but plywood is rated for various spans “for best performance.” Look on the plywood sheets and they have a grade stamp, like 24/16. The 24 means a maximum span of 24” on the roof and the 16 represents a maximum span of 16” on the floor. (They don’t rate plywood for wall coverings.) I’d use 3/8” on walls up to 24” on center spans in your case, because appearance is not important. 

Btw, span is based on the exterior layer installed perpendicular to supports.