pines de vcc y gnd del pic16f877a
Doesn't F877 have a MCLRB pin? You have to tie this pin high (to VCC) or else the part is in shutdown and won't run.
If you saw the IC get hot, that means you were running a lot of current somewhere you usually don't want to - this can happen if Vcc and GND are backwards, or if any pin is higher than Vcc. It shouldn't happen by applying 2.5v to an ADC port when in shutdown, but I can't be sure.
The problem is that sometimes if you drive current the wrong direction, you burn some junctions inside the chip, and it never works right again - that was the point of throw's warning. But as the chips are expensive, maybe it's worth looking into whether it's still salvageable..
I would put a current meter in series with the 5v supply - it should not register more than a few mA when the chip is running, and very small currents when the chip is in shutdown. Also check the 2.5v source - it should not be giving ANY current to the ADC. One of these is your culprit. If the ADC pin is sinking current from the 2.5v, it is probably burned. Maybe try the next one over just for debugging, but all in all this chip is injured and should probably be replaced.
First, apply only VCC and GND. Make sure current into Vcc not too high, and check for clocking on OSC2 - I assume you're using external osc or a crystal, this will be your best check if the IC is alive.
If the chip is up, and clocking, your program should be running. Now you can try applying analog voltages to the ADC's.
good luck!