This is a common question. The simple answer is No.
House coal and smokeless coal should not be burned in a wood burner. The main difference between a wood burner and a multi-fuel stove is the design. Wood burners are designed to burn Logs.
In a wood burner the wood is placed on the floor since wood needs air from above to initiate combustion. Coal on the other hand, requires air from above and below. Multi-fuel stoves have a raised grate where fuels are placed.
Therefore, I would recommend one to only burn wood or wood based products like briquettes in a wood burner.