For Day of the Doctor specifically, Eccleston was asked to return and ultimately decided to decline. In an interview with SFX magazine, quoted by GiantFreakinRobot.com, Moffat said:
’I had an initial contact with Chris and, in a very amiable and gentlemanly way, he didn’t feel that he could come back to it,’ he says. Moffat continues, ‘There wasn’t any big fuss about it — I had a couple of meetings with him, and he was perfectly pleasant, and indeed quite enthused about the show, but he just doesn’t do that, it’s just not him.'
In general, it's hard to say. Eccleston did not leave the show on good terms, according to the Bad Wilf podcast:
“I left Doctor Who because I could not get along with the senior people. I left because of politics. I did not see eye-to-eye with them. I didn’t agree with the way things were being run. I didn’t like the culture that had grown up around the series. So I left, I felt, over a principle."
Eccleston's been very tight-lipped about his departure in general, so it's hard to say which "senior people" he's talking about, but bad blood is as likely a factor as anything else.
Moffat's comment in my first quote seems to indicate that Eccleston doesn't like to go go back to projects once he's moved on from them, which is another plausible explanation.
That possibility is given credence by Eccleston himself, who, when asked if he'd be returning for Day of the Doctor, reportedly said:
"No, never bathe in the same river twice."
Omegacron raises an interesting point in comments: why not just CGI Eccleston's face into the show, since the BBC owns the rights to the Ninth Doctor's appearance?
For one thing, they sort of did. The last scene of Day of the Doctor, pictured below, features the likenesses of all previous Doctors including both John Hurt and Christopher Eccleston:
And of course he appeared in archive footage during the big "All the Doctors fly in to save Gallifrey" scene.
Moffat commented on this in Doctor Who Magazine, as well as the decision to not show John Hurt regenerating into Eccleston, as BBC America reports:
“It was one thing to include [him] among all the other archive Doctors, as they flew in to save the day,” he explained, “in fact, it would have been disgraceful to have left anyone out – but placing him in that [regeneration] scene might have given the impression he’d actually turned up for filming, which would have been crossing the line.”
He added: “not taking part in the 50th was a difficult decision for Chris, taken after a lot of thought and with great courtesy, and not respecting his wishes would have been grossly unprofessional and disrespectful to a good man and a great Doctor.