It's been established that opening the TARDIS doors while time-travelling has unpleasant and unpredictable consequences. This happens in a First Doctor story, Planet of Giants, and causes the TARDIS crew to live the plot of Honey, I Shrunk the Kids.
It happens again (on purpose this time) in the Second Doctor story The Enemy of the World where it causes decompression so strong that the TARDIS crew are nearly sucked out. This actually happens to the main baddie, and the Doctor describes his fate as "unpleasant".
Similarly, it appears to be impossible for the TARDIS to dematerialize with the doors open; in "The Husbands of River Song", the Doctor says (emphasis mine):
DOctor: Maybe the engines are interfering with themselves. Wild theory, but what if this machine had certain safeguards. For instance, maybe it can't take off when a life form registers as being both inside and outside at the same time?
River: Head and body.
Doctor: Which would mean, and again, I'm just, I'm just wildly theorising here, that the door would not engage properly.
River: Of course. It can't seal the real-time envelope.
Doctor Who Christmas 2015: "The Husbands of River Song"
Although not explicitly stated in the episode, presumably the doors also do not engage (and thus the TARDIS can't take off) when the doors are open.
So the only way the TARDIS would have been able to let Clara in would be to first stop de-materializing. The show never explains why he doesn't do this, although personally I like tilley31's answer in the question comments: the TARDIS was running an emergency program, and the Doctor's instructions overrode everything else.
It's also possible that this was a safety feature of TARDISes: you can't abort a de-materialization once it's begun, or once it's de-materialized a certain amount. We have no evidence to support that, but it's a fair guess.