In general, the sign of a convexity adjustment is decidedly positive, as per Johan Jensen. Complications may arise when the convex pay-off function is intertwined with other issues, like measure changes, curve slope, discounting vs projection curves, etc.
If you define it as the adjustment that needs to be made to the expected value when valueing the index of interest under a measure other than the index's natural measure, then the convexity adjustment can be of either sign. This is a meaningful definition for a convexity adjustment. It is decidedly not necessarily positive in general!