AxiomGreg:
The OP wrote "Well my question is the following: when I set the output pins to HIGH to turn off the led, wouldn't that cause a reverse voltage on the led?"He is correct here, yet nobody originally acknowledged that. Whether it actually matters in practice is another question, but I noticed that the initial responses seemed to deny that reverse biasing even occurs in this particular situation.
Well, as was discussed in the earlier posts, to avoid reverse biasing, you would want to use a driver that only pulls cathodes low and anodes high, such as the MAX7221.
AxiomGreg:
For some LEDs, reverse biasing may be a problem according to the data sheet. I have included a link below to a data sheet which warns that "Reverse biasing the dot matrix is not recommended, will cause damage to the leds." (the reverse voltage here is 3v). So I believe the best answer as to whether or not it is an issue is "it depends on what the data sheet indicates" and since the OP didn't specify the leds, we can't be certain if there is a problem or not.
I do wonder if these LEDs from 2009 are somewhat obsolete! That limitation is unusual.
AxiomGreg:
My diagram was "lifted" from elsewhere, I did not mean to specify any actual pin numbers, but your warning is well advised. I should made a few more edits before posting so as not to mislead.
The thing is, an Arduino can drive individual segments and can be programmed to pull in one direction only, defaulting to INPUT when not driving, but not drive the common return anyway.
AxiomGreg:
Having said all of the above, I have used Charlieplexing before, and it does require reverse biasing if you max out the capability (ie. use all n*(n-1) possible LEDs). But, when I do, I make sure to be under spec.
Charlieplexing is always dependent not only on reverse biasing, but on three times the diode voltage drop exceeding the driving voltage. 