In my New Zealand accent in both whine and whale the 'h' sound is slightly aspirated in contrast to wine and wail that do not have it at all.
It is very slight and may not be noticeable to other speakers at all but my lips are more pursed and I can feel the air pass through them.
Btw your question reminded me of this Family Guy clip, which is good for a laugh. (Here there is extra emphasis and stress placed on the h for comedic purposes)
Family Guy - Wil Wheaton
Regards
PS: Don't know how to embed videos here, anybody able to enlighten me?