Indoors or Outdoors
Barometers should only ever be placed outside if it is going to be used for measuring atmospheric pressure. If it is left indoors there are too many variables that can affect the barometer and falsify the readings, such as radiators and air conditioning.
Suitable Locations
For similar reasons, a barometer cannot simply be put up outside. There are a few rules to follow and you may have to move the barometer as the seasons change. The key is to put it in an area that does not see rapid variations in temperature. Place it in a shaded area under a tree or eve of a roof.
Unsuitable Locations
Do not put a barometer in an area where it gets direct sunlight, or where there is another factor affecting the ambient temperature. For example, if you place it near a window or conservatory the glass may magnify the heat. The same is true for air vents for the kitchen or air conditioning system.
Ornamental
Many people now display barometers indoor purely as a decorative item. Antique barometers are beautifully made pieces of technology and are commonly made out of high quality wood with brass fittings. If it is not necessary that the barometer works, then you can place it anywhere.