English is a word-order language. That means, among other things, that it's necessary to place modifiers as close as possible to the word or phrase they modify.
In the example sentences in your question, the placement of manually is important not as much for meaning as for clarity and ease of understanding.
What you want the sentence to mean -- that the verification should be manual -- is clear and instantly understood if the sentence reads
Manually verify that the table includes the configured values.
or
Verify manually that the table includes the configured values. (It's not necessary to put commas around manually in this case.)
As you correctly point out, if manually comes at the end, the meaning may be ambiguous:
Verify that the table includes the configured values manually.
In this case, however, all this misplaced adverb does is cause the reader to stop and have to think about what is being said. That's not good writing and it's annoying to the reader.
For manually to modify configured values, it would have to come immediately before configured.
Two other ways of writing the sentence are to use the adverb manually as a sentence modifier rather than merely a verb modifier:
Manually, verify that the table includes the configured values.
or
Verify that the table includes the configured values, manually.
Both are idiomatic and grammatical and semantically the same. And unless the context calls for one of them (because of the style and structure of other similar sentences), neither is optimal:
Manually verify that the table includes the configured values.
seems to me to be the best choice.
