Scenario in which the opamp Input offset voltage is more than inverting input?
The input offset is little to do with the closed loop gain so long as the input common mode is within the specification of the opamp.  It is true, some opamps do do odd things if the CM range is exceeded.

As your input is in mV and if you are intending this to be a digital signal then the eventual output is 3.3V, you can bias the input to ensure the CM limits are not violated, regardless of the input offset.

There are opamp topologies that allow you to tune out the offset, though any offset can drift with time and temperature so this might not be what you want.

Chopper amplifiers or self calibrating opamps are used where you want to completely eliminate offsets.

There are many opamps available these days that have offsets in microvolts, so you should have no trouble sourcing one.

It's impossible to give an exact answer as you don't specify what type of sensor you are using; what output it produces, what range of pressures you want to measure, what accuracy and if you want only positive pressures. Because of this I will answer the main part of your question, 
"what will happen if the input offset exceeds the magnitude of the signal"

The simple answer is that the output will saturate, clamp itself to the maximum (or minimum) output and stay there.  Some opamps behave oddly if the output saturates, e.g. suddenly gain a 180 degree phase shift (a positive clamp suddenly becomes a negative one or visa versa).

My suggestion would be to choose your opamp carefully to avoid the offset.

The suggestion of a differential opamp configuration offered by others is misleading, you only need this if your sensor produces a signal that can be either positive or negative without biasing.

If you want best accuracy you may want to consider an instrumentation amplifier, this allows you to have much higher gains, so less gain stages and will have a low input offset.