Both of these words are quite unusual in their spelling.
busy has an /ĭ/ sound for the U. This is the only base word we know of where U says /ĭ/ in a stressed syllable. You will also hear it in derivatives of busy, such as business.
Note that when it is in an unstressed syllable, U does sometimes say a schwa sound that is similar to /ĭ/: for example, minute. Any vowel may say a schwa sound in an unstressed syllable. In English, the schwa sound is usually similar to short /ŭ/, but it sounds more like /ĭ/ in certain words.
bury is the only word we are aware of where U-R says short /ĕ/ followed by /r/.
As I think about how the different sounds in bury are formed in the mouth, I wonder if the way English speakers said the word might have shifted over time to make it easier to pronounce. /b/ is pronounced in the front of the mouth, and fairly high in the mouth. When we say /r/ before /ē/, it is also pronounced high in the mouth and fairly forward; it is in a different place in the mouth than where we say /er/ (as in burr). In the context of these sounds, it seems to me that /ĕ/ feels much easier to pronounce than /er/ does.
These are some of the 2% of words that contain some sort of exception to regular spelling patterns. The spelling rules and phonograms taught by LOE rules explain 98% of English words.