No. You do not need to refrigerate your E3 probiotics.
Why refrigeration is not necessary: The probiotics are freeze dried which takes the active water content down to a level that isn't conducive for microbial growth, so there is no need for refrigeration. Some companies will recommend refrigeration anyway with similar products, but it can be counter productive due to how hygroscopic these lyophilized powders are. If the bottle was taken from the refrigerator and opened in a warm, humid room it would immediately wick moisture from the air being at a much lower temp and possibly compromise the long term stability of the product.