Many offices have a water dispenser (with hot water spout) or a hot water tap next to a sink. Is this hot tap Kosher? Assume for a moment that someone uses the tap to dispense hot water into a cup of non-Kosher instant soup or non-Kosher instant hot chocolate. Does the steam from the cup make the tap non-Kosher?
The Star-K advises as follows:
In general, these are permissible. If one witnessed a co-worker submerging the spout into non-kosher food, the urn should not be used. In many cases, however, one does not actually witness the co-worker placing the spout in the bowl but is concerned that it is a possibility. In those cases, one should make an assessment of the likelihood of this happening. If the spout is sufficiently far away from the tabletop so it would be unnecessary to bring the bowl up to the spout, one should simply wipe the outside of the spout and run a little water through it before filling a cup. If the spout is so close to the tabletop that a bowl would not fit easily without touching the spout, one should not use the urn. Regarding the steam rising from the bowl, for various halachic reasons, the spout remains kosher. In such a case, one need only wipe the outside of the spout and dispense some hot water that should be discarded and then take hot water or coffee for use.
You may use the water from the tap.
The Rema writes (YD 105:3):
אסור לערות מכלי שיש בו שומן כשר לנר דולק שיש בו חלב או שומן איסור ובדיעבד אין לחוש.
It is forbidden to pour from a utensil that contains kosher fats into a lit candle that contains non-kosher fats, but after the fact one should not be concerned about it.
This seems to be similar to your case. However, there are at least two independent reasons why your case can be permitted:
The Rema says that bedi'eved it is permitted, and taking water after the other person used it for non-kosher is a case of bedi'eved, not lechatchila. My basis for saying this is the Shach in the se'if before (se'if katan 5) who defines lechatchila and bedi'eved this way. Meaning that lechatchila in this type of situation means take precautions as if doing x will cause problems, but once x is done we don't see any issur whatsoever, and the thing may be consumed lechatchila.
More importantly, the details of the Rema's case are very significant - the hot item; the candle is on bottom. Therefore the basic rules of tata'ah gavar (for explanation see here starting from 3) are able to come into play and say that non-kosher flavor transfers. On the other hand your case is exactly the opposite - the bottom; the contents of the person's cup are cold (certainly not kli rishon hot). Therefore the Halachic rules of flavor transference say that there is nothing to be concerned about.
I've elaborated extensively on these points in this blog post.