The Doctor got the tea from somewhere, anywhere, but...
not on Skaro. He teleported.
After all, Missy has just explained (and demonstrated) this to Clara:
using the Daleks weapons to charge the teleporter device that she's using. And the Doctor has a similar one. Missy explained an event in which the Doctor had used this device to help explain to Clara how they managed to teleport rather than die.
It seemed to me pretty obvious that this was the trick that the Doctor had used to not get shot by the Daleks. We know that every time the Doctor deals with the Daleks, he misleads them (and Missy does the same in this episode, and multiple companions have similarly done the same). He plays mind games with them which always manages to prevent them from outright shooting him.
In fact, Davros...
may not even have a force field. There's absolutely no evidence of that in the episode, and only The Doctor makes mention of it.* Why would Davros even need a force field? After all, he also explains that is the one flaw of the Daleks, respect for their father. The Daleks would never even think to shoot Davros. An anti-Dalek force field doesn't even make sense for Davros' chair.
Later in the episode, however, there is a force field in play...
...but it is that of the TARDIS and has nothing to do with Davros' chair.
Although, in regards to Mac Cooper's comment, I wouldn't be entirely surprised to see another reference to the scene in which the source of the tea is actually revealed.
I'd also concede that...
...since The Doctor has ditched the screwdriver in favor of Sonic Sunglasses... anything is probably possible now. I cannot remember whether he had his sunglasses during this particular scene, but we may find out in the near future that this may have had something to do with it.
* To be clear, I mean to say that The Doctor is the only one to mention it in this episode. Having thought about it, I realize I'm certainly not an expert in all things Who, and I haven't seen all of the episodes from the original series, so perhaps this force field IS a thing that can be confirmed from the old episodes. But more than anything, we know we can't take The Doctor's word for it, certainly if the only mention of it is in this particular episode.