Actually, a good question. To me, the "but" on that sign seems perfectly natural, but when I stop and think about it. It is a bit odd, because it doesn't really connect the two parts of the sentence. To answer your question, I think that there is something implied that is not made explicit. Perhaps: We apologize for the inconvenience. (We would really like everything to be in perfect working order), BUT the elevator is currently out of order. I will be curious to see other answers to this.

"but" is used to introduce a reason why you couldn't do something, OR to explain and give the excuse why something happened. In your example, "We apologize for the inconvenience BUT the elevator is currently out of order", the "but" introduces the excuse for the inconvenience. The excuse it introduces is that the elevator is out of order. A few other examples. I would have gone to work today, but I was sick. The "but" introduces the excuse why I did not go to work. The excuse was I was sick. I apologise for not going to your party, but I was very tired. Again, the "but" introduces the reason/excuse for not going to the party. The reason or excuse was that I was tired.

BUT= we apologise BUT we have tom say something impolite, impolite being breaking the bad news about the elevator being out of order. It's a grammatical construct e.g. "I hate to tell you the bad news BUT we have a pop-exam today".

it is used to introduce a phrase or clause contrasting with what has already been mentioned. ie. "he stumbled but didn't fall".