I cannot find any record of a connection between the two. Perhaps the plots between the two have less in common than you remember?
In the Impossible Astronaut the Dr discovers the Silence, an alien race, have been living amongst and influencing humanity without our knowledge or being able to see them.
That isn't quite correct - everybody can see the Silence - they just instantly forget them.
In They Live, the aliens' motivation is to take Earth for themselves and live on it. The Silence were only influencing key people on Earth to one single end - to assure the Doctor's death.
In the John Carpenter film They Live the hero John Nada discovers that aliens live amongst humanity, controlling and influencing them to do as they wish through subliminal messages and control of a small number of collaborators. He is only able to see them when he wears a special pair of sunglasses as they are able to use conditioning and a host of other techniques to make humanity ignore them or see them as human.
The Silence did not live amongst humans. They did not take human form, nor integrate into human society.
At the end of They Live the hero destroys the signal generator hiding the aliens from sight and humanity is them able to see them and fight them. In Day of the Moon the conclusion to the 2 part opener of Dr Who the Dr uses the moon landing and the words of one of the aliens to force humanity to see them.
In They Live, a broadcast was being constantly sent out to mask who the aliens were. It was broadcast from a TV station, but it did not come through the TV - you didn't have to be watching TV to be influenced by it. Nada's mission is to destroy the mast and stop the signal.
In Doctor Who, there is no constant broadcast. The Doctor hijacked a TV signal, not to stop it, but to use it to broadcast a message. This was only received by people if they were watching TV, and the Doctor chose the moon landing because it was being watched by so many people on Earth.
I agree that They Live is an influential film - if anything, They Live has more in common with The Matrix... earth taken over by non-humans... nobody can see the world how it really is except a group of rebels... the protagonist encounters the rebels and is invited to join them... protagonist is enabled to see what everybody else can see and is compelled to join the fight.
When you consider the finer details, the only similarities between They Live and Day of the Moon are an alien race hiding on earth and the Doctor revealing who they really are. That could describe any number of Doctor Who stories from any era.