Nothing; the aircraft continues on the current active leg.
The figures describing the desired track, change to true and have 'T' annotated against them in the FMC.
All you need to do is to re-align the heading bug against the lubber-line (if that's what you like to do)
The only confusion can arise if you are displaying VOR needles/course bars, as they will point in the wrong direction, i.e. they will be 'stuck' to the magnetic bearing and therefore will move with the compass card. ADF needles will stay on on the correct relative bearing (i.e. they will not move) but will now indicate a true bearing on the RMI.
Eck