Is the tabernacle candle always lit and burning?

Even at night when no one is around?

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.

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