Will these LEDs work for throwies?
You need a series resistor to limit the LED current. You can find out how to calculate its value here. In this case your problem is that you have little overhead combined with large tolerances in voltages. The voltage difference between Vbat and Vled can vary between 0.1V and 1.25V. A new battery should give you over 3V; I've measured voltages of 3.4V on CR2032s. OTOH, 2.5V for a red LED is rather high, so I would use the 1.85V for my calculation (even that is a bit high!). So, taking 3.1V for the battery and 1.85V for the LED you need a series resistor of (3.1 - 1.85)V / 20 mA = 62 ohm. A E12 value of 56 ohm will do fine.  

edit
You probably have seen the battery directly placed between the legs of the LED, no resistor, but this will drain the battery fast.