Which of the following pair of words are not homonyms:

The correct answer is Option (3) → burn, burden

Homonyms, in the wide sense used in such questions, are words that sound the same (and may or may not be spelt the same) but have different meanings. Burn (to be on fire, to damage by fire) and burden (a heavy load) neither sound alike nor share a meaning, so they are not homonyms.
Example: Keep the old letters, for they are no burden; do not burn them.

Why the Other Options are Incorrect?

Option 1: cite, site — Incorrect. Both are pronounced alike: cite means to quote as evidence; site means a place where something is built or happens.
Example: She cited a report in her essay; the new school will come up on this site.

Option 2: pallet, palate — Incorrect. Both are pronounced alike: a pallet is a wooden platform for goods (or a straw bed); the palate is the roof of the mouth, or one's sense of taste.
Example: The boxes were stacked on a pallet; the spicy dish did not suit his palate.

Option 4: hear, here — Incorrect. Both are pronounced alike: hear means to perceive sound; here means in this place.
Example: I can hear you well from here.