Is there a general logic as to how the syllable "ate" is modified in different circumstances?
The notion that literate being a descriptor and obliterate being a verb, has merit in the matter of pronunciation of the last syllable.
My first thought was that "literate" had an unstressed final syllable because it had entered English earlier and had been subjected to the truncation of the end sounds that happened to many words in the late 1400's into the 1500's. That may have been the case. The OED will not date the word's entry into English but other sources suggest the 1400's. The word may well have evolved from LIT TER AH teh to LIT TER uht in speech over the last 500 years.
There is a general agreement that "obliterate" came into English a couple of centuries later. But no principle can really be established by that. "Commiserate" a verb, and "commensurate", a descriptor, arrived at about the same time. The verb has the final sound of obliterate, and the descriptor the sound of literate. The word "certificate" is more to the point. As a verb its last syllable is stressed like "obliterate", as a noun it has the final syllable of "literate" (OED).
If a principle can be stated, it would be that a final "-ate" in a verb is stressed and in polysyllabic words that are not verbs it is unstressed. (verb examples, monosyllabic: mate, date, and rate all of which may be used as non-verbs.... polysyllabic verbs: consecrate, congratulate, deflate....polysyllabic non-verbs: confederate. degenerate, legitimate, illiterate).
No principle in English is really rigid. However it does seem that "-ate" at the end of a verb wants to be stressed. It would sound strange to a speaker of Modern English to hear someone yell "Obliteruht it!"