Is it harmful to the Buddha when we request he or she to continue teaching instead of entering parinirvana?
As I pointed out in my answer to "are earthworms our mothers?"  these kind of questions often assume incorrect frame of reference. The reason we request Buddha (or our local teacher as Buddha's substitute) to continue teaching instead of entering parinirvana, is not because it can literally make Buddha (or teacher) change their mind and stay. The idea is to create a mental context in which A) the teacher feels motivated to continue teaching because he sees the teaching is in demand, B) the students stay aware of the fact that existence of the teaching in this world depends solely on Buddha and then generations of teachers passing it to one another, and now to the students, and C) the student clearly realizes that the teacher could very well leave this place but is instead making effort to stay and teach solely out of compassion for the students. When this context is established the teaching can happen effectively, because both the sides will engage actively and with no resistance.

Granted these are entry-level practices, but they are very important to establish as foundation. Otherwise what happens if you skip straight to advanced teaching, the student may get as arrogant as to metaphorically grab the teacher by the hair and swing around. Needless to say, it hurts the student first and foremost.