I have not used birch syrup, yet...
I have however used maple syrup and maple sap. The trouble with using these ingredients and it is similar with birch, is that they are highly fermentable and therefore not much flavor will be left behind when the yeast gets through with them. Even if you use a lot it will not be very apparent especially if you are making a roast centered beer such as a porter. The roast will obliterate the birch character in my opinion. The smoke phenols will also mask it somewhat as well.
I have not yet had the pleasure of trying birch syrup. Does it have a wintergreen character to it like you would get from using birch twigs?
Your best chance of getting the flavor to show trough is to add it at the end of the boil at flameout so as to drive off the least aromatics. Also you can add some to the fermenter once fermentation slows down, it will still ferment but you won't lose as much aromatics through the airlock. Finally you could use it to bottle condition (or keg) as your priming sugar. Amounts? Well, with maple I have used as much as a half gallon $$$. I make some syrup at home so it costs time rather than dollars. I have also used partially boiled sap for the brewing liquor. That was in a barley wine so the maple character really had no hope!
Other things you can consider in the future are using the birch sap in place of water for your liquor. Possibly using some twigs as well as they use in making birch beer. Good luck and have fun with it. It may take a few batches to get the effect you want but as long as you have as source it is fun to try.