How can I troubleshoot stuck/flipped bits in a serial connection?
Suggest checking whether the problem recurs at differing bps ("baud") rates:


If the problem goes away at low bps rates then it seems an "analog" problem such as grounding continuity issue (maybe the chip has multiple ground pins and one's open?) or ringing ('scope it out, though it seems unlikely at RS-232 rates ) or charge accumulation leaked from the high voltage needed for driving the VFD
If the problem occurs regardless of the bps rate, then perhaps there's a logic, framing (wants 1-and-half or 2 stop bits, computer set for one?) or clocking issue issue.


Other things to look for:


Where does the clock source for the UART come from?  Is there an external clock or crystal?  Perhaps the board was designed with an intentionally off-frequency clocking which happened to be convenient for the phone's CPU's clock divider, allowing a very cheap, but non-standard serial bps rate via "software UART"
Presumably the board has a crystal for its clock.  Try padding it with a few pF in parallel to see if such varying of the chip's clock rate straightens up the framing.
When the rational fails, try the moderately irrational: 

o Another computer?  Another USB cable?

o If you're ready to give up, presuming the chip has an external crystal, driving one or the other crystal pins through a high series impedance (to avoid chip damage) such as perhaps a 100K ohm resistor from a variable frequency sine wave generator at various frequencies slightly above and below the crystal's frequency.  The idea here is to override the crystal's frequency and take control of the chip's clock, varying to see if this straightens up the framing and bit decoding.
Might the circuit supply a clock-out somewhere for serial rate clocking?