It's not clear from your description what the inputs are doing. Do you mean that if you attempt to apply a logic level high (+5V or +3.3V, perhaps) from the PIC's output to the input, then the voltage measured at the input drops to 2.5 V? That would mean something relatively bad is happening, since the output pin voltage won't drop unless a large current is being drawn, probably more than the rated absolute max.

So your inputs seem to be drawing significant current. Inputs shouldn't usually do that; so it would be a good idea to get as much documentation as possible, and see if you can figure out how you're supposed to be driving them. Perhaps they really do need the high current, and you need some kind of high current driver, or perhaps you're driving them with an incorrect voltage.

You might also try measuring the voltage from the PIC's ground pin to each of the two wires that your relay shorts when things work correctly; that might give some insight as to the correct expected input voltage.

It may be that you can drive your input with some type of open drain / open collector output, or with some other circuit. But the solid-state device that behaves most like a relay is an optocoupler, for example a phototransistor, like

http://cq.cx/interface.pl#4 ,

or a photo-FET like the PhotoMOS family (e.g., AQY284EH) from Panasonic. Optocouplers transmit information from the input to the output using light, not electricity; so they can work when there's no electrical connection between the two sides (which is not the case here, since you said they both ran off the same power supply), or when there's an electrical connection with some undesirable voltage between the two sides (which could be the case here).