Is there any risk of illness from microwaving and eating a packaged frozen dinner?

Yes. A pathogen called Listeria monocytogenes can contaminate pre-cooked foods, including frozen entrées.
 
The meat, dairy, and cheese contained in frozen entrées are particularly susceptible, though other items, including pre-cooked vegetables, also may harbor the pathogen. During processing, listeria is killed by cooking and, in some cases, an additional pasteurization step is added after the product is cooked. However, contamination can occur after these procedures but before the product is packaged.
 
Freezing the entrées doesn't kill listeria. It merely inhibits bacterial growth. Therefore, be sure to follow heating instructions for frozen entrées. Proper heating will kill listeria.
 
Depending on how and how long they are stored, leftovers may be subject to spoilage bacteria and mold growth. Spoilage bacteria probably won't make you sick, but mold can. Moldy leftovers should be discarded.
 
Check the label for common allergens such as dairy, egg, soy, and seafood products. The obvious bears repeating: If you or a family member are allergic to a food or food ingredient, don't eat the food. Freezing or cooking the food won't alter the outcome.
 
Source(s):
USDA; FDA "Listeria Monocytogenes Risk Assessment--Questions and Answers"
About.com "Food Poisoning and Food Spoilage"
Foodallergy.org "Common Food Allergens"