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If it's an absolute must, I would invest in a high quality air purifier and place it right next to wherever it's being burned. (Along with moving the bird cage as far away as possible). Be as careful as possible. Good luck!
Truthfully they aren't very good for you or your fireplace. When they burn they leave a coating inside the chimney that builds up to the point of becoming a fire hazard if not cleaned regularly.
Having said that it's not so important what you burn in a modern efficient fireplace because the fire gases never mix with the room air (two chamber system).
If it is just a regular firebox with a chimney then there is quite a bit more risk because the two airs mix. Also for example if you light a fire in a cold fireplace/chimney there is a good chance of blow back, filling the room with smoke.
If your fireplace is like this you can burn anything.