What exactly are you referring to as your phone charger. The charging circuit for the phone is internal of the phone itself. The small block that you plug into the wall is a power adapter, not the charger. And the cable you plug into the block and then the phone is just the connection from the adapter to the charging port on the phone.
If you smelled something burning, then it is very possible the adapter failed, or there was some debris in the charging port that caused a short. If Apple replaced the battery, then bring it back and advise them it will not charge. There is a warranty on the battery replacement. It is also possible there is something wrong with the charging circuit in the phone, and the phone would need to be replaced. Apple should be able to diagnose that problem.