Since most modern era dice are cellulose acetate, the problems of deterioration over time are similar to that for photographic film. Here is an excerpt from an article on storage of motion picture film. Basically... keep the items cold and dry, and ventilated to dissipate the acetic acid they generate. A sealed medicine bottle would not seem to be the best way to store them.

A link to the whole article is below.

The deterioration of both cellulose nitrate and cellulose acetate negatives is highly dependent on temperature and relative humidity. Ideally, to minimize decomposition, negatives should be stored in a freezer. At freezing temperatures the natural decomposition of cellulose nitrate and acetate is slowed down. The cold storage of small numbers of negatives can be done easily, but the cost and inconvenience of freezing a large collection can be prohibitive. However, cold storage is predicted to extend the life of acetate negatives by a factor of ten or more. (See "Image Permanence Institute Storage Guide for Acetate Film" by James Reilly.)

A good resource on cold storage is Henry Wilhelm's "Permanence & Care of Color Photographs" (1993), Grinnell, Iowa: Preservation Publishing Co. Chapter 19. A fairly inexpensive cold storage unit is a large commercial freezer which should defrost automatically. Storage materials designed to preserve photographic materials are archival boxes placed in polypropylene bags and then sealed with humidity control cards. This allows the stored items to warm at room temperature safely (8–12 hours should be sufficient) and can be easily accessed. The Safe Care® Archive Freezer Kits are available at Metal Edge, Inc. (800) 862-2228.

A less costly option for storage would provide a controlled environment with the constant temperature at 68° F (20° C), and relative humidity between 20% and 30%. Rapid changes in temperature and humidity will hasten deterioration. A dark and well ventilated area around the negatives will allow the gases to dissipate.