Depending on the kind of tire pressure monitoring you have it could set your warning light off more often. Anything - like the stability control, ABS, and some TPM systems - that depend on measuring relative wheel speed can be affected. But, like the monkey said, if it isn't acting up now, it probably won't act up with new front tires vs worn front tires.
What is TPMS & How Does it Work? | Bridgestone Tires