If the tree bark is fresh off the logs and not dried out then the tree bark might cause some creosote buildup.
But if you let the tree bark sit and season for at least 6 months prior to burning it then the tree barks should not cause much if any creosote buildup in the chimney.
I burn tree bark that has seasoned during the shoulder seasons such as now in October as I find it wasteful to just throw it out.
The tree bark burns fast but it does provide some heat so I use it in my wood stove and never had an issue of building up creosote from it.
The most common cause of creosote buildup in a chimney is not burning the wood stove or fireplace hot enough to get the temperature up in the flu.
When the temperature of the flu is too cool and not hot enough then condensation can buildup in the chimney which leads to creosote.
Also burning wood that is not fully cured or seasoned such as green wood or wood that still contains some moisture can be a cause of creosote buildup in a chimney.
Chimney fires are caused by too much creosote buildup that then gets burnt and caught on fire from the heat and fire in the chimney.
You should make sure you burn the wood stove or fireplace hot enough to keep condensation and creosote from building up.
Also have your chimney swept every year before each burning season starts because it's best to have a clean chimney and it's cheaper than losing your life or losing your home to a house fire.