Overwhelmingly, we do not pronounce the first d. A recent facebook poll I conducted had 53 people saying Wenzday versus only nine who said Wed'nzday.
Wednesday was originally "Woden's Day", the day of the Germanic god Woden. The Germanic tribes got the idea from the Romans, who had named their days after the five planets that they knew at the time, Mercury being one of them. The Norse equivalent of the trickster god Mercury, who also accompanied the dead to Hades, was Odin, or in Anglo-Saxon, Woden, so Mercury's Day became Wodnesday.
|Woden. You don't want to mispronounce this guy's name.|
Likewise, a variant of "Wodnesdei" was "Wednesdei".
The string of consonants d-n-z-d is quite hard to say, so already by the 13th century there is evidence of a form where the consonants are switched around (a phenomenon called metathesis), so that the word was spelled "Wendesdei". From there, it was but a short step to lose the first d in the pronunciation altogether, and this was indeed happening by the late 14th century.
So, as Wodnesleie ended up as Wensley, so Wodnesdei ended up, at least in the pronunciation, as "Wensday". In spite of that, the "Wednesday" spelling, which was only one of many possible competitors for the prize, established itself by the end of the 17th century.
One cannot argue that because "Wednesday" is spelled the way it is, it is "wrong" or "lazy" to pronounce it "Wensday". Lord knows we have plenty of unpronounced letters in English!
However, some people (a very small minority, according to my survey), do pronounce both d's in Wednesday. This pronunciation survived particularly in some varieties of Scots English and in northern English speech. Interestingly, the very first person who replied "Wed'nz day" to my poll was born in Scotland. The others may have acquired it from growing up in a particular Scottish-influenced area, of which there are many in Canada.
For the origins of the other days of the week, click here.
For more on metathesis, see this post and this one.