I have already learned from more than one source, that "vowels あいうえお are never pronounced as diphthongs". I understand that, and I know how to say "青い". However the way I hear it, there seems to be an exception to this rule - namely two vowel combinations:
1. いう is pronounced as ゆ or ゆう (long vowel)
2. いえ is pronounced as "ye" in yesterday
(言う and いいえ for example.)
On the other hand combinations of a consonant with two vowels are pronounced the way I would expect. For example 消える is read "ki-e-ru" instead of "kye-ru".
Am I deaf or there is some other explanation?

The verb 言う is usually pronounced ゆう (and you even hear ゆって instead of いって as the -te form).

Obviously there's a lot of variation in speakers but at least in "standard" Tokyo pronunciation, いえ is pronounced normally as い + え. You may hear a slight "y" sound in there just because of the way the two vowels sound together, but I don't think it's as pronounced as an actual y-glide (or even the "y" sound in something like 千円).

I think the warning that "vowels あいうえお are never pronounced as diphthongs" is more likely to be referring to the pronunciation of individual vowels specifically, due to the native English speakers' tendency to turn simple vowels into diphthongs (e.g. o -> [oʊ̯], e -> [eɪ̯], etc.). The pronunciation of Japanese vowel phonemes in combination with each other is a separate issue.