Wood swells when wet and shrinks as it dries. The warping may be due to dampness as might the drawer sticking.
Put it in a warm dry place and wait for it to dry out before attempting any remedial work.
Candle wax rubbed on the bearing surfaces of the drawers sometimes eases sticky movement.
You may be tempted to shave or sand off the lower edges of the sticking drawers but this should be a last resort - difficult to put it back on again if you remove too much.
If the carcass is very badly warped it may need some dismantling and reassembly - glue and cramps.
Treating with teak oil will restore the finish if it's bare wood. The dirt of ages - everyday use - may be removed with ultra fine (0000 grade) wire wool and white spirit
Peccavi
January 2009