We have a log burner, its amazing - we don't get any smoke in the house. It all goes up the chimney. Now we don't have our chimney lined, we fitted the burner ourselves before the building regs 2010. My DP now fits these professionally on occasion and they have to be signed off by biulding control.
The building regs stipulate that the chimney should be lined with a steel liner and they can last anything from ten to 25 years.
In your shoes i would get your chimney examined by a qualified chimney sweep/hetas engineer and also the buner, to make sure it is drawing properly (taking the air from the room and not returning carbon monoxide) and that there are no leaks. There are a series of simple tests that can establish this along with the soundness of the chimney.
You should have a carbon monoxide detector fitted, we have one both down and upstairs but in all honesty so long as their is adequate ventilation and draw on the chimey.
With regards to ventilation, depending on the size of the burner you will need a permanently open vent in the room (they are quite big) to comply with building regulation. In an old, less airtight house it isn't so important but you obviously need enough air to fuel the fire and also prevent fumes coming back into the room.
When you moved into the house were there building reg certificates available for the burner - there should be a certificate of competence installed somewhere prominent by the fitter which will state that either independent building inspector has inspected the installation or the fitter was registered with hetas.
If this isn't the case - it might be worth getting building control to inspect it, your local council will advise you about this. It costs probably about £500 depending on your area.
If everything is as it should be, it is perfectly safe, but do make sure that you have it inspected.