I got an idea about connecting serial (RS-232) GSM modem to TI-83+ (and using it to send SMS, for example).
My guess is that it should be possible, since calc has serial interface (GND, Rx, Tx) and serial modem would be used (unless modem requires some more wires).
What do you say?

What kind of baud would the modem expect from the calculator and how accurate does it have to be?

The first problem is that the calculator only runs at 6MHz. This means your theoretical max baud would be 285Kb/s, the next would be 230, then, 206, 200, and then they get to more manageable increments. But, I guess if your expected baud is under 200Kb/s, you'd be ok.

However, a more severe problem is the lack of exact timing on the 83+, if your modem doesn't really care, it shouldn't be a problem, but no two calculators run at the same frequency. I can't give you exact numbers on the 83+ (because there's no timing system), but on the 84+ line, I've seen calculators all the way from 14.5MHz all the way to 17MHz. Since you are relying on the cpu to clock your data, you won't be able to get very precise timings.

However, if you sort out both of these issues, the rest should be relatively easy.

The 83+ (running at exactly 6MHz) should be able to pull off 113,210 baud, 115,380 baud, or 117,650 baud. Hopefully the modem is either lenient or uses a much lower baud than that. Also, to muddle the problem, we can probably assume cpu speeds from 5.8MHz to 6.8MHz which puts that middle baud in the 111,540 - 130,770 range.

Which makes the arduino look like a good choice