Is the tabernacle candle always lit and burning?

[John1990]

Even at night when no one is around?

 

[Fr.Charles_Grondin]

Yes:

Canon 940 A special lamp which indicates and honors the presence of Christ is to shine continuously before a tabernacle in which the Most Holy Eucharist is reserved.

On the worship of the Eucharist, 25 May 1967

1. **The Means of Indicating the Presence of the Blessed Sacrament in the Tabernacle
   **
   …According to the traditional practice, a lamp should burn continually near the tabernacle as a sign of the honor paid to the Lord.

Holy Communion and Worship of the Eucharist outside Mass
11 …According to traditional usage, an oil lamp or lamp with a wax candle is to burn constantly near the tabernacle as a sign of the honor shown to the Lord.

Usually it is a candle inside a glass container placed inside a glass container securely hanging or standing near the tabernacle. This manner essentially eliminates any possibility of an issue with fire/accident.