On my BC2500, when I stick in a tape, the unit will play for 1-2 seconds then stops. If I press TAPE again it will play for another 1-2 seconds, then stops. The transport controls (ff, rew, step, play, direct track selection) do not engage at all.

The transport will work continuously however if there is no tape in the compartment

Is this another belt issue?

Looks like it. I have changed some belts but I must say that it's a lot of work to replace. You almost have to dismantle the whole unit including the deck itself. The "old" belt is sometimes heavily disintegrated and looks like tar wrapped around the motor pulley. Messy job to clean. 

Some new info: I was wondering if this could be an electronics issue, since the FF, PLAY, and REW keys do not work at all, only TAPE --

I started the deck with no tape in the well. It will run for about 10-16 seconds then stops. 

If I started it again and pressed on the tape sensor switch (middle one, angle shaped, tells the unit that a tape is loaded) the deck would then stop within a second or two.

The unit basically wants to run with no tape in it, but *not* run with a tape in it... Is there some "control board" thing that could have gone bad? The unit is from around 1996, and the tape unit has only 2-3 hours on it, max.

Try playing a 30 second tape if you have one.  This could tell you whether it is electronics or drag from the tape.  Some old tapes have a ton of drag on them.