The IRS won't influence the FMC calculated position,maybe it was just a coincidence
I believe thats incorrect.
In a dual IRS configuration, the FMC uses 5 positions to arrive at the
system position.
You've got
left IRS position,
right IRS position,
radio position,
best position and
system position.
The best position is heavily biased towards the radio position, which of course may create problems if its inaccurate.
The system position is updated using the computed best position which is the navigation position used by LNAV.
This is all from Bill Bulfer's FMC Guide.
IMHO then, any malfunctioning IRS has the potential to cause a noticeable shift in FMC computed position.