Is there a limit on how much current a pin can sink?
am I going to fry my board?


Pretty likely outcome. Here's why:

The Arduino Uno uses the atMEGA328 microcontroller, which has an absolute maximum rating of 40 mA source or sink per GPIO. Also, the total current through the supply or ground rails (i.e. the total of all current OP wants the GPIO pins to sink, or source) is rated to a maximum of 150 200 mA. 

Realistically, the current graphs in the atMEGA328 datasheet do not even go all the way to 40 mA, hence it is safest to restrict the current to perhaps 20 mA total per GPIO.

In other words, whether a GPIO is used as source or sink, the limits remain pretty low, and in practice, should be kept much lower than the absolute maximum ratings. Even below absolute maximum, running a microcontroller to too high a temperature reduces its life expectancy.

Solution: 

Use either BJTs, MOSFETs, or some driver IC such as the ULN2003 to actually drive the current through the LEDs.