It's not impossible that you have a hardware failure, but I'd suggest trying other things before going the repair route. I'm assuming that you're using "known good" MIDI cables.

Did this difficulty occur recently? Had you been able to previously send and receive SysEx messages without problem? If so, have you made any hardware or software changes since the time when things worked?

Early DX7 units sent "Active sensing" (FEh)messages all the time, which can add clutter to the MIDI data stream. (I believe that a replacement ROM allowed that to be controlled.) You might want to see if it can be turned off on your DX7. If not, perhaps it can be filtered out by your software.

If you already have backups of any "user" material in the DX7, you might try doing a factory reset on it in case something you haven't considered is misconfigured.