if transformer secondary remains open, and primary is excited by sinusoidal source will there be heating produced?
If secondary coil of a transformer is left opened, and primary is
  excited by sinusoidal current source, will there be heating produced
  in the core of secondary ?


As both the secondary winding and primary winding must share the magnetic core of the transformer (or it isn't a transformer), this question boils down to examining what heat is produced in the core when there is no-load on the secondary.

The answer of course has to be "yes" unless the transformer is a perfect device with no losses. The losses in the situation described are called "iron" losses and this is a general heading for: -


Hysteresis losses (energy lost in reversing magnetic domains)
Eddy current losses in the core or core laminates (energy lost due to circulating currents in the non-zero resistances of the laminates.


These energy losses produce heat. Lower down the order of things but certainly not insignificant in other transformer applications the following should also be noted: -


If the transformer is a high frequency device then dielectric losses
should also be included.
If the transformer has low magnetization inductance (primary
inductance), the current that flows will be significant enough to
generate \$I^2R\$ losses in the primary winding. Not to be confused with
\$I^2R\$ losses due to load on the secondary.