Almost no one does it except professors and cosmopolitans. Though some books will say that "what" should be pronounced "hwutt" and not "wutt", is it really recommended for us, the common folk, to espouse this pronunciation; or would we sound affected in doing so?
As others have noted, w(h)ether you should pronounce "wh" as /w/ or /ʍ/ depends on what the prevailing regional accent does. The big exception to this is singing; it pays to be fussy about pronunciation when you sing, even if you wouldn't in normal speech, because it helps the words to come through the texture.
Incidentally, you shouldn't think of /ʍ/ as sounding like "hw". /ʍ/ is an unvoiced aspirant made with the "w" mouth shape, not an /h/ followed by a /w/.
This question explains about its origin and usage. Basically, to sum up, "wh" was originally pronounced "hw" due to its origin.
who - /huː/ (Old English hwā)
whom - /huːm/ (Old English hwǣm)
whole - /hoʊl/ (Old English hāl—cf. 'hale')
As you can see, it was originally spelt "hw", changed to "wh", and now, its pronunciation is slowly changing to "w" due to a phenomenon called the "wine-whine merger":
The wine–whine merger is a merger by which voiceless /hw/ is reduced to voiced /w/. It has occurred historically in the dialects of the great majority of English speakers. The resulting /w/ is generally pronounced [w], but sometimes [hw̥]; this may be hypercorrection. The merger is essentially complete in England, Wales, the West Indies, Australia, New Zealand, and South Africa, and is widespread in the United States and Canada. In accents with the merger, pairs like wine/whine, wet/whet, weather/whether, wail/whale, Wales/whales, wear/where, witch/which etc. are homophonous.
As you can see by this image, that "wh" is stilled pronounced "hw" in quite a bit of USA.
It is true that "wh" is increasingly being pronounced "w", but that doesn't mean that pronouncing it as "hw" is "affected", or snobbish.
So, I wouldn't be bothered using either, as both pronunciations are actually acceptable.
Listen to this and decide for yourself if it sounds affected.
The goal of oral English is to be understood clearly in the initial iteration. If one wants to communicate 'where' something is located, aspirating the word clearly differentiates it from the word 'wear', as in reference to apparel. Ambiguity, repetition, and wasted time can thus be avoided. It is a matter of logic.
If you fail to pronounce the combined "wh" sound, how is a person to know if you mean "whale" or "wail", "where" or "wear", whether" or weather, etc. I am a singer, and diction is extremely important or those listening won't know which word you mean. It sounds like mush coming out. Do we really want to start speaking sloppy? I know "I" strive to make the distinction.
I think dropping the "h" is an embarrassing mistake.