Should I use MLCC or Tantalum capacitors near the board's power input?
For larger values Tantalums can be cheaper and more space-efficient that the same effective capacitance in ceramics.  Depending on the dielectric, ceramics can have very poor voltage and temperature coefficients so a 10uF 10V ceramic cap may only have 1-2uF of capacitance with 10V DC on it.

However, Tantalums are prone to failure from high ripple current and overvoltage.  Ceramics are much more rugged.  If you have a buck converter (DC-DC) be very careful with tantalums.  Polymer tantalums are best and be sure to not exceed the ripple current rating or the voltage rating even for a short period.