I am a native speaker of a very general General American and I would never omit the /t/ in "next year". I don't insert a schwa either. Many people say /nɛks jɪər/ and I would certainly understand it. However, you should be aware that it's a dispreferred linguistic variant and might sound slightly "uneducated" to some people, if they noticed (it's a very minor detail.)

"Don't ask, don't tell" has two glottal stops and one strong aspirated t. Omitting those also sounds colloquial to me.

I am a native speaker of a very general General American and I would never omit the /t/ in "next year". I don't insert a schwa either. Many people say /nɛks jɪər/ and I would certainly understand it. However, you should be aware that it's not standard and may sound too casual. Don't worry if you can't pronounce it that way, though; most English speakers wouldn't notice (it's a very minor detail).

"Don't ask, don't tell" has two glottal stops (in "don't") and one strong aspirated t (in "tell"). Omitting some of those also sounds colloquial to me.