Wow! Thank you everyone for all the feedback. This was my first post here and I didn't expect the topic to be this popular.
Thank you Tooki for the excellent link to the reference on old school LEDs. It was very illuminating (pun intended.)

I think I may indeed have those very old 50ma indicator LEDs. I think that for several reasons:
- The first Hayes Smartmodem was released in 1981, so assuming the design process began a year or two earlier, that puts it in correct the timeframe for the 50ma indicator LEDs described in the reference you linked to. As far as I know they used the same LEDs for all their future Smartmodems, so my mid-80s modems would probably still have the same ones.
- In the reference you listed, it says that the later, higher-efficiency LEDs would change the shade of red if they were in a clear package, depending on the current. The LEDs I have are in a clear package and seem to produce the exact same shade regardless of how much current I put through them, they just get brighter or dimmer. I've tested them all the way up to 78ma, with no discernible change in color shade. They always look red, not orange.
- I've tested them all the way up to 78ma! I would expect a typical 20ma LED to burn out quickly at that high of a current? I left one running at 45ma for days, and it seemed fine and did not get warm to the touch. Even at 78ma I was able to run it for a few hours. At that point it finally did start to get warm and the light from it began to look orange, so it would seem that 78ma is past its breaking point.
To those that asked for measurements in the original circuit: again, I can't measure them in the original circuit, because I don't have a power supply for it. Also, being an electronics newb, I have a pretty limited bench supply... it's just a converted ATX power supply (I know, I know...) so I can't easily produce the 13V the modem required. And again these were "parts" modems that I bought cheap, so I don't believe them to be operable anyway.
For those that asked for other measurements: my multimeter set to diode modes measures their Vf as 1.76. Running them at 5V on my breadboard with various resistors I get the following:
- 22ohms = 78ma(1)
- 47ohms = 45ma(2)
- 100ohms = 25ma (3)
- 200ohms = 14ma (3)
- 1Kohms = 3ma (4)
- 10Kohms = 300ua (5)
(1): I decided to sacrifice one of them in the name of science: it ran for several hours at 78ma before finally showing signs of burning out.
(2): I ran one at 45ma for several days with no signs of harm. This was the only value that looked to me like a reasonable output level for an old-school indicator LED. Yes, I know that LEDs of that vintage were very dim. (I'm old enough to have used these types of modems back in the day.) Still, at any less current than this they are VERY dim, as in barely visible kind of dim. That was what inspired me to start this post in the first place, rather than just running them at 20ma to be safe.
(3): They were visible at this value, but really seemed quite dim, even for 1980's indicator LEDs.
(4): *barely* visible, just a tiny pinpoint of light that I needed to turn the room lights down to even see.
(5): No visible light that I could see.
To those that suggested using modern LEDs or cutting up the old ones to use as light pipes... yeah, that would probably be the sensible thing to do, but these old Hayes modems did have a very unique look to them. It's hard to describe... sort of a glowing pinpoint of light that I think gets magnified by the clear housing. which has a domed lens in it. I just don't think it would look the same with modern LEDs.
Also, I agree with you 100% Tooki on modern indicator LEDs. I guess I'm just old, but I don't get it. I have a blue power LED on my TV that I had to cover with tape because it is so bright that I couldn't even watch the TV without constantly being distracted by the glare. What were they thinking? Now that I'm finally getting a bit more into electronics I may start swapping some of them out.)

I've finished prototyping the project on a breadboard (with modern LEDs) and programming the Pi, and it's pretty awesome if I do say so myself. I even hooked up the modem's speaker to the Pi's audio output, and I have it play mp3s of appropriate call progress and carrier sounds at the appropriate times. It really does feel like I'm dialing into a BBS via a phone line in the pre-Internet days.

To those that asked for photos, here you go!
These are the actual LEDs we've been talking about:

And here are the high-res photos of the modem circuit boards that were requested. Because they're so big I'll just provide links instead of inline images. The LEDs were soldered on over on the right-hand side:
1200 bps modem - component side:
http://www.nullmodem.org/eevblog-photos/1200-front.jpg1200 bps modem - back side:
http://www.nullmodem.org/eevblog-photos/1200-back.jpg1200 bps modem - daughter board component side:
http://www.nullmodem.org/eevblog-photos/1200-daughter-front.jpg1200 bps modem - daughter board back side:
http://www.nullmodem.org/eevblog-photos/1200-daughter-back.jpg9600 bps modem - component side:
http://www.nullmodem.org/eevblog-photos/9600-front.jpg9600 bps modem - back side:
http://www.nullmodem.org/eevblog-photos/9600-back.jpg