Well, not directly, so don't worry about it.
The point here is that you made a choice that you will remember and notice it later on.
More information with SPOILERS, read only AFTER FINISHING THE GAME:
Thanks to this choice you will almost immediately notice that the girl at the end is not Elizabeth since she doesn't have the brooch you selected for her. It was a nice touch by the game creators.
I think I got the answer, and if you think about it, it's quite straightforward. It isn't Booker, or any metaphysical representation of his psyche.
The lighthouse serves as an entrance to Columbia, and as we see at the beginning of the game, used by pilgrims and newcomers that have been accepted into the city. Such a place would have a keeper, a guardian. There are religious motifs all over the place, because as any good Columbian, he'd be a believer. There are charts in the lighthouse that show where the city is going to be depending on the time of the year, and the pattern it follows.
And to activate the chair that ejects you into the city, you need a password, the bell combination, that only said lighthouse guardian would know. Now, in the Lutece labs, you find a picture of the lighthouse with a red circle marked around it and the words "ONLY ONE OBSTACLE" next to it. Their plan was to send Booker through the lighthouse, and the only obstacle was the man guarding it himself.
So that's it. The Lutece have shown to be quite cold in their behavior and not care much about other people's lives when they get between them and their experiments. Maybe they didn't do the deed themselves, they might've just hired another nameless thug to do the dirty job.
But basically: It's nobody but the Lighthouse keeper, who's been tortured and murdered in order to get the means for Booker to get into the city. They might've left the "Don't dissapoint us" message to add color or feed him the idea that he's working for some shady criminal organization.
Best Answer
Probably, you're referring to this city in the distance:
Effectively, the lights may look like the city is on fire. If you see the same scene from the top of the lighthouse though you'll have a different perspective:
As you can see from here, it just looks like these are cities in the distance, with their pretty city lights.
So my guess is that there's nothing on fire, it's just that the lights aren't optimised for looking through the lighthouse windows.