What should be done to bed pillows to reduce allergies, asthma and dermatitis?

Hypoallergenic. Buy hypoallergenic ones, and wash the case once a week or so using a detergent that does not bother you.
Allergy proof pillow. In addition to dr. Walsh's suggestions, if you are dust mite allergic, you should get mite-proof pillow and mattress encasing. They will keep dust mites from aggravating your respiratory allergies during the hopefully 8 hours you lie in bed each day. You can purchase them from bed, bath & beyond and various allergy supply web sites. Microfiber fabric is the most comfortable and most expensive.
Eucalyotus oil. Use hypo allergic cotton. Kill dust mite with eucalyptus oil in the wash.