What is the requirement for having a hazardous material inventory?

Any examples are for illustrative purposes only.  
Any hazardous material or waste that is regulated by local, state, or federal law (including OSHA, EPA, DOT, etc.) are required to be part of your organization's current inventory of hazardous materials and waste. This inventory may either be consolidated into one document or decentralized.  Consumer products (such as turpentine, gasoline or white out) that are used in a workplace in such a way that the duration and frequency of use are the same as that of a consumer, are not required to be included in the hazardous material and waste inventory. However, it is the responsibility of the employer to make the determination for their workplace by assessing the exposure potential of the consumer products that staff may encounter and ensuring that the frequency and duration of use are not greater than that of normal consumer use.  

A good rule-of-thumb would be, for a given product, review the Safety Data Sheet (prior MSDS) and determine if the organization's method of use could result in adverse exposure.  If the SDS contains any storage or usage warnings, like special storage, special criteria for the use environment, critical emergency actions to take if exposed, etc. then those products should be included in the hazardous materials inventory.  Hazardous wastes are typically tracked by manifest, and that acts as an inventory.