In order to generate a tube system of low resistance, sieve elements lose most of their cellular components such as the nucleus, cytoskeleton, ribosomes, tonoplast etc.
Mature sieve elements contain structural phloem specific proteins (P-proteins), mitochondria, ER, and sieve elements plastids.
In plants, sieve tube cells lose their nuclei at maturity (just like our RBCs), and thus lose their chromosomal DNA, but they don't lose their mitochondria.
In eukaryotic cells Wikipedia has divided extrachromosomal DNA into three parts based on where it is seen (I guess):
Sieve tube cells have mitochondrial extrachromosomal DNA at maturity, but I'm not sure about the existence of extrachromosomal circular DNA in these circumstances.
Do sieve tube cells have any extrachromosomal circular DNA at maturity?