Why are Greek letters pronounced incorrectly in scientific English?

Some of the factors include:

• Ancient Greek pronunciation was, it is believed, significantly different to the modern language. How do we know? Well, that's a fascinating topic, but there are various clues, including the way animal noises were rendered (suggesting that the ancient "beta" was closer to "b" than "v", because sheep's bleating sounds more akin to "baa, baa" than "vee, vee"!). Wikipedia's article on Ancient Greek phonology has more details.
• The pronunciations have come to English via other languages (French/Latin), which have exerted their own influence.
• The pronunciations received from them went through the Great Vowel Shift, which probably explains the "pee"-versus-"pie" for "pi" aspect.

Also note that the pronunciation in English is not itself consistent: in my experience most UK speakers say "beet-a" for beta, whereas many US speakers pronounce the same letter as "bate-a".

The scientific pronunciation is based on the Classical Greek pronunciation, not the Modern Greek pronunciation. In Classical Greek:

• η is [e] ("ay"), not [i] ("ee")
• υ is [y] ("yoo", approximately) or [u] ("oo"), not [i] ("ee")
• αυ is [au] (the vowel in "out"), not [av] or [af]
• β is [b], not [v]
• δ is [d], not [ð] (the "th" in "this")

The only cases where the English scientific rendition is "wrong" is for ι, which was [jota] even in Classical times, χ which is usually pronounced as [k], since the velar fricative [x] doesn't exist in English, and ψ and ξ which are simplified to [s] at the beginning of words since English doesn't allow [s] to be the second element in word-initial clusters. These represent adaptations to English phonology.

The Greek language came to English and other languages through Latin, and at that time, Greek letters had a different pronunciation than today, plus the changes they went through when passing from one language to another.
Something similar happens with Pekin/Beijing, or why we name Persians to people that call themselves Farsi

I'm not that familiar with the history of Greek pronunciation, it's quite probable that the modern pronunciations of these letters in Greek have drifted from the past pronunciation -- before they were used in English.

I would guess the pronunciation was influenced by Latin, as well. Greek words used in English normally pass through Latin. It's probable Latin influenced the pronunciation of β, certainly.