This question actually has a few answers… depending on the quality of your BBQ and your climate.
First up, some higher-end BBQs are built to withstand temperature and weather changes. Obviously BBQs are designed to be outside and some higher-end more expensive BBQs are designed to last four seasons outside without too much protection. Covering these BBQs can be a good idea but if you forget for a day, week or month you will likely just need to give it an extra wipe down before using it next.
The next thing to consider is the climate. If you experience humid climates (or rather damp climates) it may not be a good idea to cover your BBQ. The reason being that the humidity (damp air) can get trapped under the cover. Being trapped it is less likely to dry out and can lead to corrosion. If you do choose to use a cover in these conditions it is a good idea to choose a cover with some air vents and periodically take it off to give your BBQ a wipe down.
In coastal climates, erratic weather and salty air can cause corrosion. It is a good idea to keep BBQs in these areas covered, particularly if you are aware of any heavy rain or other ‘erratic’ weather. Obviously, with the salt being in the air, there is a good chance that it will still make its way under the cover to giving your BBQ a good wipe with a damp cloth fairly regularly is a good idea.
Being mindful of the weather, how it affects your BBQ and covering it when you need to will help extend the life of your BBQ and ensure that you enjoy plenty of barbecues!