In Star Trek: Discovery, Michael Burnham was sentenced to life imprisonment. Given that she is serving on Discovery, does this mean that her crime is forgiven?
Not at all. Burnham has not been pardoned. In fact, S1E5, "Choose Your Pain" makes it clear that she's basically been pressed into service under Star Fleet regulation 139.82, which allows conscription in a time of war. While she no longer seems to be confined to quarters when off duty (for example, nobody thought it was odd she attended the party in S1E7, "Magic To Make the Sanest Man Go Mad"), she still has no rank--strictly speaking she has to take orders even from Cadet Tilly right now. She has merely been remanded to the Discovery's custody for the time being.
In S1E8, "Si Vis Pacem, Para Bellum", and S1E9, "Into the Forest I Go", Burnham makes it clear that she fully expects to be remanded back to prison when the war ends.