Why does this component on my Arduino UNO heat up?
According to the Uno rev 3 schematic that part is an LMV358 dual op-amp, it drives the mosfet power selection switch and also drives the "L" LED. https://upload.wikimedia.org/wikipedia/commons/7/7d/Arduino_Uno_Upload.png   

It you continue using the board there is a chance the chip will eventually burn even further and potentially cause other problems, in an extreme case the chip might blow its top throwing tiny shrapnel bits around.  

The op-amp section that drives the LED has a series 1k resistor, so unless there is a short ahead of the resistor that shouldn't be the problem, but does the LED still flash at times? The other op-amp section drives the gate of a mosfet that selects the main system power source (usb or external).    

Mosfets, especially discrete parts, can be very susceptible to damaged from a static zap.  During winter months low indoor humidity can greatly increase the potential for static problems.  Since one pin of the mosfet connects directly to the usb 5v there is a chance that just the act of plugging and unplugging the usb might have  caused a static zap to the Uno board.  

If the mosfet's gate became shorted to either the drain or source pin that could easily cause heating of the op-amp chip, (the op-amp's output would then be shorted to one of the power supply lines).  Check if the mosfet is also getting hot.  The mosfet is the small 3 pin device just to the right of the fuse (which is just below the usb connector in the Uno picture you uploaded).

If you suspect the mosfet is defective you could try removing and/or replacing it, (the mosfet type is listed as an FDN340P.)  But if you do take that repair route keep in mind two things, without the mosfet the Uno will only operate when power is supplied from the DC jack.  Another issue is that in removing the mosfet there is a chance that just in that process you could damage the part, so if you really want to remove/replace the mosfet do use static protection, (a static wrist strap and even connect the Uno ground to the same ground as the strap). But before doing any component work do check for other obvious problems like shorts or bent pins, also be sure your not shorting something out with other I/O lines too.  

If you do decide to try a repair it may be best to just go ahead and replace both the mosfet and op-amp.