Can unconnected inputs make an IC get warm?
Yes, CMOS circuits can get hot when there are floating inputs. You should always connect unused CMOS input pins to a defined voltage, usually GND or Vdd, unless the datasheet tells you otherwise (see also the end of this answer and Michael's answer). If a pin could be configured as either input or output and you are not sure which it will be, then you could place a resistor between the pin and GND/Vdd.

If you leave pins unconnected, they are said to "float" and have an unspecified voltage. That voltage can be from induction on the package leads, leakage currents inside or outside the package, static discharge, etc. The key point is that you don't know the voltage at the gates of the input transistors to which the pin is connected (signal A in the CMOS inverter below).



In the worst case, this undefined voltage will be somewhere between "high" and "low", so that both transistors are conductive at the same time. Thus, a high current (several 10-100 mA) flows through the transistors from Vdd to GND (Vss), thereby generating heat and possibly destroying the chip.



Some ICs have special circuits at their input pins to prevent this from happening. This circuit is typically called bus-holder or bus-keeper, but can also be found under other names like pad-keeper (e.g. i.MX processors). It is essentially a buffer (two inverters in series) and a large resistor connected to the input pin. This ensures that the input pin is always driven to either high or low when nothing else is driving it.



Image sources: Wikimedia, public domain.