-리 is a literary sentence ending similar in meaning to -ㄹ 것이다 or -ㄹ 거야.
It conveys a mixture of one's intention and future prediction. It is too literary for use in speech, but quite common in song lyrics and in writing. It gives a feeling of a resolve, hope, and wishful belief.
There is also a longer form -리라, which is also common.
For example,
- 오늘을 잊지 않으리(라) = I won't forget today.
- 이 모든 게 헛되지는 않으리(라) = (I know/believe) All this won't go to waste.
So 빛나리 is like "They/it will shine" or "They shall shine".