Why is 'diphthong' pronounced the way it is?

We break diphthong into syllables differently than the Greeks did. We break it diph-thong, whereas etymologically it is di-phthong. Because there's a consonant on the end of the first syllable, it's natural for English speakers to pronounce it with a "short i", /ɪ/.

The same thing happens with diptych, whose etymology is di+ptykha, where ptykha means folds.

In words from Greek or Latin, a single vowel letter before a consonant cluster that cannot¹ occur at the start of a word tends to take its "short"² pronunciation. The consonant cluster in the middle of "diphthong" cannot come at the start of a word (whether you pronounce it as /fθ/ or as /pθ/), so the "i" in the first syllable is pronounced as /ɪ/.

The converse is not true: before a consonant cluster that can start a word, a single vowel letter may either be "short" or "long". For example, "diploid" is pronounced with /ɪ/ even though there are words that start with /pl/.

As Peter Shor mentioned, the criterion can also be formulated in terms of syllabification. An "open" syllable ends in a vowel, and a "closed" syllable ends in a consonant. In general, we can say that non-final closed syllables in English tend not to have "long vowel" sounds, although that isn't a firm rule of English phonotactics: there are words like "paraleipsis", where the penultimate syllable is closed and pronounced /laɪp/. This is why I included spelling in the rule that I gave at the start of this answer: it would be possible to have a word pronounced /ˈdaɪfθɒŋ/ or /ˈdaɪpθɒŋ/, but it wouldn't be expected to be spelled with diphth-.

There are different approaches to syllabification, but one commonly used criterion says that a consonant cluster can start a syllable only if it can start a word. This criterion implies that consonant clusters that cannot¹ occur at the start of a word like /fθ/ always cause the preceding syllable to be closed. Consonant clusters that can occur at the start of a word, like /pl/, would typically be expected to form a syllable onset when the occur word-medially, but some people would argue that the use of /ɪ/ in the first syllable of "diploid" implies the syllabification dip.loid. It's unclear why diploid would be syllabified differently from diglot.

1. According to the phonotactic rules for typical English words. Exceptions like some people's pronunciation of "phthalate" do not count. Incidentally, some people pronounce words like "psychic" with /ps/, but /ps/ also does not count as "a consonant cluster that can occur at the start of a word" for the purposes of this rule.

2. "Short" in the phonological, not the phonetic sense: short a e i o u are the phonemes /æ ɛ ɪ ɒ ʌ/. (In American English, /ɑ/ is used instead of /ɒ/.) An alternative term that has been used is "lax".