Not usually, no.

The true Jasmines (genus Jasminum) don't, at least not any usually grown in the UK.

The unrelated Star Jasmine (Trachylospermum jasminoides) which is evergereen does sometimes produce reddish foliage as the leaves age and when the temperature drops. We had a large one in a container that was becoming root bound and starved and it went a wonderful reddish colour, not normally so extensive of course in normal conditions.