What makes old spice flammable?

Answer 1

After looking at the ingredients in the roll-on variety it's because of the variety of organic compounds1 within. (Almost all listed ingreients are in this category) In the spray variety it's likely because both of that and the high surface area2 of the mist, perhaps the propellants even contribute.

1. Organic compounds are compounds consiting only of carbon, hydrogen, and oxygen (Such as sugars, alcohols, and natural gas) They are typically combustible and produce carbon dioxide and water as products of combustion.

2. Surface area is the exposed portion of any material. Dividing a substance into smaller pieces and pulling those pieces away from each other increases its surface area. Increasing the exposed portion increases the amount that can react with the neighboring oxygen at any time. With enough surface area flourcan detonate a concrete silo!