I wouldn't consider it safe to burn/heat anything scented in a house that has birds in it. By the time they start showing effects of the scented items it might be too late to help them.
I'd also be very leery of putting out poison because if one of your birds finds one and eats it it will cause you heartbreak.
Something that was recommended to me by a coworker (apart from the standard traps, etc.) was to put dryer sheets in those corners/places that aren't used or opened very often.
When we first moved into the house we're in now, we found the mice were getting into the kitchen d4rawers and the the bakeware drawer under the oven. So we cleaned/disinfected everything out, washed everything, and filled in holes with that spray foam filler. We found that the mice were coming in through the heat exchanges in the floor so we put gutter grid over the hole under the exchange cover. Then we put unscented dryer sheets in the back of all the kitchen drawers and lower cabinets as well as the bakeware drawer.
Strange as it may seem, the stuff does seem to work ...