"선생님께서 식당에 갔어요" 반말(casual speech) and 존대말(polite speech) refer to the politeness, usually expressed by the ending forms. For example, 갔습니다 and 갔어요 are 존대말, whereas 갔다 and 갔어 are 반말. So the sentence above is in 존대말 form (but maybe you were talking about as it relates to the treatment of 선생님). This politeness expressed by the ending forms is directed to the listener. The other kind of politeness is with respect to a person mentioned in the sentence, 선생님 in this case. This is expressed by using honorific forms for the words (noun, verbs, particles) that refer to the person. In this example, 께서 is an honorific particle as distinguished from the plain 이. On the other hand 갔어요 is a plain form (for 선생님) , so there is a mismatch which makes the phrase a bit unnatural. 선생님께서 식당에 가셨어요 would be both more polite and more natural. This politeness of the sentence and honorific treatment of a person mentioned in it are mutually independent. So there can be these combinations (using the informal endings 어 and 어요 only): 1. 선생님이 식당에 갔어. (the most casual speech between friends) 2. 선생님께서 식당에 가셨어. (between friends, but still following etiquette) 3. 선생님이 식당에 갔어요. (a pupil to someone older, out of etiquette w.r.t 선생님) 4. 선생님께서 식당에 가셨어요. (a pupil to someone older, in good etiquette) * -이 ... 가셨어(요) and -께서 ... 갔어(요) are less natural combinations but are still heard fairly often. * 선생님 is also an honorific noun (님 suffix), but it's become a set word so we don't normally use 선생 by itself. As for your question, you were probably asking whether 이 should be used after 선생님 (not 가 since 님 has 받침). That will make it pair with 갔어요 better (but not in good etiquette), but as mentioned, it is not absolutely required. Also, in fast speeches the particle 이 or 께서 is often dropped, so this issue doesn't come up too often.

께서...가셨어요. '갔어요'는 반말이 아닌데요. '-요'가 붙었으니까요. 반말은 '갔어'.