Should one refrain from speaking or interrupt the recitation of morning and evening remembrances (adhkar)?

There is no harm in interrupting the recitation of adhkar (remembrances) with incidental speech, especially when needed, and then returning to complete them. It may even be recommended to interrupt for matters such as: returning a salam (greeting), saying "Yarhamukallah" to someone who sneezes, responding to the adhan (call to prayer), removing an evil, guiding to a good deed, answering someone seeking guidance, or when overcome by drowsiness.