The pronunciation of "ei" and "ou" varies whether you are pronouncing it carefully or not. Regarding the pronunciation of "ei" in "teinei", in normal conversations, the majority of the Japanese speakers would probably pronounce it more like "eh" So, it would be "teh neh" However, newscasters, actors and actresses, voice actors, and in formal speech, people might pronounce it more like the English letter "A" as in "A, B, C, D, E, F, G..." Also if you ask a Japanese person to pronounce this word alone, s/he might pronounce it as "ていねい(teinei)" because it is written "ていねい" and the word is out of the context. Just like if I ask a native English speaker how to pronounce the word "best," s/he would probably say "best." But if I ask the same person "Who is he?" then, his/her reply might be "Oh he is my bes friend" without "t" in the word "best". But it is never pronounced as "i" as in the English greeting "Hi!" or "I am" or "iPhone" because that would be "tainai" and "tainai" is a completely different thing.

Like Mitch said, the way -ei and -ou words are pronounced depends on how carefully you wish to enunciate it, particularly on its own. On its own, the word 丁寧, when trying to pronounce it carefully, is pronounced ていねい (with the "t" from "take" and the "ay" from "say," and "nay") but in actual conversations words such as this (in my experience they all follow this rule, for example ケイタイ、両方、工場、etc.) the first vowel is simply extended. In other words, ていねい becomes てーねー (with the "te" from "television" (but slightly longer, "teelevision") and the "ne" from "next" (but again a little longer, "neext")) and the same rule generally applies for -ou words as well - 工場 (こうじょう) has two -ou sounds that are technically pronounced with the "o" from "company" and the "u" from "ultra" but in actual conversations it is usually pronounced こーじょー。 Like Mitch said, we do this in English too. "Best friend" becomes "bes fren" in actual conversations. "Actual" (act-you-all) becomes "akshul" and "momentarily" (mow-men-tare-ily) becomes "momentearlee" and so on and so on. In short, you don't need to worry about enunciating it 100% accurately, because conversational simplifications like this happen in all languages. If some confusion occurs (because of homophones, perhaps) enunciating it more carefully might not work and you're better off using it in a familiar sentence (in English, if someone doesn't understand the difference between "no" and "know" you could say "the "no" in "no thanks" or "the "know" in "I know that already.")